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S\«s ra H : * i ; d ! ‘ ea Pe thlay Aa, yy avwity : Poe : ap ew tet 1" rit: i ‘Pip ies rt Fate Os Reb LA PAA F vA Me ye ee i* j i y* uv 4 "= j " or i i ral , d x 4 : & y . ; ‘ . s ; re 4 re | ’ cay 4 7 4 ¥ q : i ta 4 4 we , b ‘ i 4 3 * . “ r a J ‘ i rt i) i) +) 4 > bea < | ' a nits ‘} i . ‘ ' rey i | i re ‘ eas ; oy f ‘ , We be? a | ‘ : . i ti . é ‘ ; ‘ « he jy q + } S , Ma 4 x Tels " ‘ net {cea r ¥ bE 4 * oa ta “a } q hie eet os ; ier dere Mise S4 ; vA! t Prt hs thd Brice oe worse 7 eas TH 4 -$fhss wists F ' : y \* , 2 TF ia ‘ey a i hy ‘ 5 - “. ny ’ 7 * : rents fivch werd Lfyiaathits ral A rhea to} Ste eat 5 ’ Hee CAN ‘ 4 ff . 4 rs Ta a? a eho Oa , rt dh tage hy “iy * ‘ srt tuated wore eS he Pl Yet 7 bi Hig ; @7' Ay $e ’ ta hit mein 1p Ge l-heqed wu ttle Hit) fab yo! it Tenia I a i Mt ‘itt hat fy i] ; st Hea Hea J ihe sein ar i) righ A } abe Nabe hy eal mh She ie forse ee a ab a AAS hahaa Tt tie aa ae ey ae; Py hirs be » ae shad Bey 44 AP Way sity aie Ne 1 edae aha L dubehy Hee ni of Aa? ili Be pyph lee toh ge jhe te bieie oot bday INS nt kt pet ene fide bale ithe Ht j vey fbb ipa ps iN ¢ 0h 4} ¢ Ay erie yy Heit oo wt fh ry it Als phaadl hae od 1 tibee Ler alle ha he wpe 14) 4 ’ Uh ta ie > t we Mbt ie "ye Wid hte ano i oh ea rit Pipa ior' iby at Ht , Aa grid bey +irbe > : 1227.3 THE a a 4 a SS LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. CONDUCTED BY mR OLIVER JOSHPH LODGE, D.Sc., LL-D., PRS. SIR JOSEPH JOHN THOMSON, O.M., M.A., Sc.D., LL.D., F.R.S. JOMN POLY 5. MeAs Disesek A... Gs. RICHARD TAUNTON FRANCIS AND WILLIAM FRANCIS, F.1..8. “‘ Nec aranearum sane textus ideo melior quia ex Se fila gignunt, nee noster vilior quia ex alienis libamus ut apes.” Just. Lips. Polit. lib.i. cap. 1. Not, VOL. XLIV.—SIXTH SERIES. JULY—DECEMBER 1922. LONDON: TAYLOR AND FRANCIS, RED LION COURT. FLEET STREET, SOLD BY SMITH AND SON, GLASGOW ;— HODGES, FIGGIS, AND CO., DUBLIN ; —- AND VEUVE J. BOYVEAU, PARIS, ‘‘Meditatiouss est perscrutar] occulta; contemplationis est admirari perspicua .... Admiratio generat queestionem, queestio investigationem, investigatio inventionem.”—Hugo de S. Victore. ——“ Our spirent venti, cur terra dehiscat, Our mare turgescat, pelago cur tantus amaror, Cur caput obscura Phoebus ferrugine condat, Quid toties diros cogat flagrare cometas, “ Quid pariat nubes, veniant cur fulmina ccelo, Quv micet igne Iris, superos quis conciat orbes Tam vario motu.” J. B. Pinelli ad Muzontum. ( ALERE SH FLAMMAM, BRASS aS CONTENTS OF VOL. XLIV. (SIXTH SERIES). . 7 NUMBER CCLIX.—JULY 1922. Page Mr. A. R. McLeod on the Unsteady Motion produced in a Uniformly Rotating Cylinder of Water by a Sudden Change in the Angular Ro em Cee ON MUAD YS 71 ocak 9 ig ask Fick scaddo Hten vig ERO NN Dr. Norman Campbell on the Elements of Geometry ............ 15 Dr. Dorothy Wrinch on the Rotation of Slightly Elastic Bodies ... 30 Mr. G. H. Henderson on the Strageling of « Particles by Matter .. 42 Dr. G. Green on Fluid Motion relative to a Rotating Harth........ 52 Prof. G. N. Antonoff on the Breaking Stress of Crystals of Rock-Salt. 62 Dr. Norman Campbell on the Measurement of Chance ............ 67 Messrs. H. T. Tizard and D, R. Pye: Experiments on the Ignition of Gases by Sudden Compression. GR VAE eye E so ete Mics 79 Mr. C. Rodgers on the Vibration and Critical Speeds of Rotors.... 122 Mr. P. Cormack on Harmonie Analysis of Motion transmitted by “So Tig SNS he NOTTS Shee Bip a cota gre ca or Pn ee Teh ae Messrs. E. W. B. Gill and J. H. Morrell on Short Electric Waves 156 CMR NEV ES etext. Molar Sus sees wh ad ave wR ee epee acne ha: 161 Sir George Greenhill on Pseudo-Regular Precession.............. 179 Dr. J. W. Nicholson on the Binding of Atoms by Electrons ...... 193 Mr. L. St. C. Broughall on Theoretical Aspects of the Neon Spectrum. 204 Dr. F. H. Newman on Absorption of Hydrogen by Hlements in the Remon Misenaroe Tbe). Fe oem. bls chats tet de eet U oliacs lao Bie asad 215 Mr. B. A. M. Cavanagh on Molecular Thermodynamics. II. ...... 226 Prof. J. G. Gray on the Calculation of @entuoids" Sv acs. hance tae 247 Messrs. A. P. H. Trivelli and L. Righter: Preliminary Investiga- tions on Silbersteia’s Quantum Theory of Photographic Lxposure. 252 Dr. L. Silberstein on a Quantum Theory of Photographic Exposure. 257 Mr. R. F. Gwyther on an Analytical Discrimination of Ilastic Sarecses Mn atl SORrU ple: Ody tire Paes. kh ecw cee tl eves 274 Mri, S. Rowell on Damped Vibrations’ . 2.6.0.0 ce Yee Fk. 284 Notices respecting New Books :— Mr. L. #. Richardson’s Weather Prediction by Numerical Ma RE ee eresigce earn ie MEMES ONT ach A sriorh Pelee aot helasin oc acaee Os 285 Proceedings of the Geological Society :— Mr. C. E. Nowill Bromehead on the Influence of Becky on FHRoEMAMeMVEGE IOUMONS ss eka gets ck ewe... LOO Intelligence and Miscellaneous Articles :— On Young’s Modulus and Poissona’s Ratio for Spruce, by Mx. H. Carrington : 1V CONTENTS OF VOL. XLIV.——-SIXTH SERIES. NUMBER CCLX.—-AUGUST. Page Mr. C. J. Smith on the Viscosity and Molecular Dimensions of Gaseous Carbon Oxysulphide. (COS) oauioc et ey. olen = eine 239 Prof. A. O. Rankine on thé Molecular Structure of Carbon Oxy- sulphide and Carbon Bisulphide = - ce . sam arene 292 Mr. F. P. Slater on the Rise of y Ray Activity of Radium Emanation. 300 Profs. J. N. Mukherjee and b. C. Papaconstantinon on an Iixperi- mental Test of Smoluchowsli’s ‘heory of the Kinetics of the: Process-of Coagulation: 22. diy os) wedge sok tee eee nee ee 305 Prof J. N, Mukherjee on the Adsorption of lois) 227.2 ae 321 Prof. W. M. Hicks on certain Assumptions in the Quantum-Orbit. Uheony ol Spectra: jv. 25 cp ao te seco sacs. 2 nto aie gemen tae eee eae 346 Mr. F. C. Toy on the Theory of the Characteristic Curve of a Photo- erapline WanuilsiOm Gi. tiaie one! « cos. Slane see eee 352 Mr. A. M. Mosharrafa on the Stark Effect for Strong Electric CNG Siig ce a aes On usuag: a eal tale ah Shae es Cae A ee, er 371 Mr. I. Takagishi on the Damping Coefficients of the Oscillations in Three-Conpled MiectriesCircuits.. 72.2. ena Sa eee 373 Prof, 8. C. Kar on the Electrodynamic Potentials of Moving CHar oes es i takes ae nod Gig wate Rae Is eee GAGE eee a te a O Mr. A. KE. Harward on the Identical Relationsin instein’s Theory. 380 My. H.S. Rowell on Energy Partition in the Double Pendulum... 882 Piof. J. S. Townsend on the Velocity of Electrons in Gases ...... 384 Prof. H. A. McTaggart on the Electrification at the Boundary between a Miquid-andia Gas .4....5 2.2 4) ee een eee 386 Prof. L. V. Ising on a Lecture-Room Demonstraticn of Atomic Models: «Plate dd.) acs ea ee eee oe 395 Mr. Hi. D. Murray on the Influence of the Size of Colloid Particles upon the Adsorption ‘of Mlectrolytes|s a... ... cn 4.2 eee 401 Notices respecting New Books:— Dr. G. Scott Robertson’s Basie Slags and Rock Phosphates... 416 NUMBER CCLXI—SEPTEMBER. Prof. Sir EK. Rutherford and Dr. J. Chadwick on the Disintegration of Klements by o Pagucles: ce cho ce sete been 417 Prof. W. L. Bragg and Messrs. R. W. James and C. H. Bosanquet onthe Distribution of Electrons around the Nucleus in the Sodium and Chlorine A toms’ gist ne mesic aces ect tere ee ee 453 Messrs. C. G. Darwin and R. H. Fowler on the Partition of Dual) 02 ae ea eM HI en rate eras ila) SU Gen ere Sat hsa’n x 450 Mr. M. EH. Belz on the Heterodyne Beat Method and some Appli- eations. to Physical Measuremientse ese ese eee 479 Mr. R. #. Gwyther on the Conditions for Elastic Equilibrium under Surface Tractions in a Uniformly Eolotropic Body............. 501 Mr. C. J. Smith on the Viscosity and Molecular Dimensions of Salles ioxide. 2. iis eae Neetgtee Saeko ean ne 508 Mr. 8. Lees on a Simple Model to Lllustrate Elastic IHysteresis..., 511 Prof. R. W. Wood on Atomic Hydrogen and the Balmer Series SI CLCIAANLLOD Gp Crete © cit eC MM MISIPREEE ont: asr bn 7°48 99 vi/c?="01 9 yy) = d1°8 99 cee Oe aoe) ne (C7 "OL ms t=1484 ,, 10 Mr. A. R. McLeod on Unsteady Motion produced Comparison with the theoretical curves (figs. 2 & 3) shows a marked departure in all cases but that of the slow speed and small cylinder. The agreement with theory improves as r—>c, but there is still a large departure at r/c="9 for the large and middle cylinders, except at the low speed. & or oF NW) eo ‘ —) ane Die: — To) on ie) -f> ee ° i) © Again, it is noticeable that as the radius of the cylinder increases, the departure from theory becomes more marked. This might be expected as there is, near the axis, relatively Jess constraint from the boundaries with the larger cylinders. Accordingly, if we suppose an eddy of this kind rotating in a lake of stationary water, and if instead of stopping the cylinder wall we annihilate it, we expect the eddy to dis- appear more quickly than if the stationary solid wall had by Change in Angular Velocity. 1a been retained, for the constraint will be still further reduced and greater irregularity is possible. It is particularly noticeable that as the constant rotation 0. of the cylinder is increased, the departure from theory becomes more marked. —S=—_ r) he 10 Only three figures were obtained for the small cylinder, because the agreement with the theoretical curves in the three cases omitted will be practically exact. The curves for the ‘‘ stopping” experiments skow a much greater departure from theory than the curves for the “starting” experiments. ‘This is due to the break-up of the regular motion, owing to instability at the fixed outer wall. Except with the high speed and the large cylinder, 12) Mr. A. R. McLeod on Unsteady Motion produced the motion on starting, on the other hand, appears to the eye to be without appreciable irregularity, and it is very striking to see the sharp dividing-line between quiescent liquid in the centre and rotating liquid on the outside. This dividing-line, represented by the steep part of the curve in the figures, slowly moves towards the centre but becomes indistinct some distance from it. It is best seen at the higher speeds when the velocity-gradient is greater. Its rate of travel depends on the value of (2. In the case of the large cylinder starting at the high speed, large secondary eddies 3:5 cm. across were often observed just inside the cylinder wall a few seconds after starting the motor. These soon died out when the velocity-gradient became less, and thereafter the motion travelled in towards by Change in Angular Velocity. 13 the centre regularly. The effect of these eddies was observed to cause a shifting of the curve for ¢/Q in the direction of a greater # for ie same value of vt/c?. In the same case, secondary eddies were observed about the “4 circle when the motion had reached the centre. Although mean velocities were recorded (median ling of the band of points), the effect is shown in fig. 17 by the wavy appearance of tl.e two upper curves. When the cylinder is stopped, the water continues to rotate until the irregular motions, generated near the cylinder wall, have had time to extend inwards. Small eddies then travel about, and the central axis of rotation wanders con- siderably and often seems to disappear temporarily amid cross-currents. The motion is very irregular except at the low speed, and even in this case some irregularity always remains. The lycopodium particles do not follow the circles for very long, and are usually moving at an angle to them. With the large cylinder at the high speed, the velocity immediately after stopping the cylinder seemed to give stability and to aid in preserving the circular character of the motion ; but when the kinetic energy had somewhat diminished, eddying became more noticeable. . On the curves mentioned in $3 in which ¢/Q is plotted against tle’, the bands of points are much narrower in the “ starting’ experiments than in the others, and determine the position of the median line easily to -001 in the value of vt/c? in most cases. For the “stopping” curves, the limit of error may be two or three times this occasionally. One noticeable effect is that the band is narrow when the velocity- gradient has a considerable value, z. e. when the curves in the figures slope steeply. In these cases considerable momentum is being transferred through the water, and there will be considerable shearing stress and vorticity, and the stability might therefore also be considerable. As soon as the velocity-gradient becomes small, the band of points broadens. For example, in the “starting” curves .the bands are some- times very narrow until the value of $/Q has risen to 0°9, when they broaden ont. Conditions seem to favour irre- gularity at the centre (axis) of the cylinder where the velocity-gradient vanishes. On the axis the stability is a minimum. In the ‘stopping’ curves the bands are nar- rower the greater the angular velocity Q, 7. e¢. the greater the vorticity of the water, especially near the cylinder wall where the instability originates. The observational curves show that viscosity alone is not 14. Unsteady Motion in a Rotating Cylinder of Water. sufficient to account for the effects, except for small values of cand Q, ¢.e. for long, narrow cylinders and: slow speeds. A little dye introduced into the rotating water shows no slons of any minute eddying or micro-turbulence ; and so we must look for currents in the water as the cause of the discrepancy, which is obviously the case in the “ stopping ” experiments. The formation of large eddies in large bodies of fluid seems to be due chiefly to the interaction of two local currents, or to low pressure caused by an obstacle or a sink, and not to the slower processes of viscosity. If we attribute the deviation from theory to an ignored increase in the kinetic viscosity v, we find that when the large cylinder is stopped at the high speed, the increase would have to be represented by a factor exceeding 10 in value nearly everywhere, while the value would lie between 50 and 100 on r/c=0°3 shortly after stopping. With the middle cylinder, stopped at the high speed, the factor has about half these values; and with the small cylinder, stopped at the same speed, the factor ranges from 1:3 to 3:0. In the starting experiments the factors are nearly unity, but they are meaningless here as the motion is not turbulent. Some earlier experiments illustrate the instability of the stopping experiments. In these an inner cylinder was rotated coaxially with a fixed outer one. As is well known, it was found that at no speed of rotation of the inner cylinder was it possible to set the water moving in circular paths, owing to the eddies which were continually thrown off. The slower the speed of rotation the more conspicuous were the eddies, especially on the borders of the outer, more slowly-moving water. Measurements of the angular velocity showed a large departure from theory, the inner parts rotating more slowly and the outer parts more rapidly than the theory indicates. The effect of the travelling eddies is thus to make the angular velocity more like that of a rigid body. When the speed was very great (2500 r.p.m.) the kinetic energy seemed to give stability to the water. A whirlpool formed next the inner cylinder, and a large oscillation was presently set up in the form of a wave with its crest along a radius of the outer cylinder and its trough on the other half of the same diameter. Some thick, very viscous oil residues, when rotated in a cylindrical tin about 15 cm. in diameter, acquired the full velocity on starting (36-r.p.m.) in something less than 4 seconds, and came to rest in the same time when the cylinder was stopped. Onlya slight displacement of the oil occurred, the surface being momentarily roughened with fine lines like cracks. Rupa? J II. The Elements of Geometry. By Norman CaMpBgé.L, Sc.D.” Summary. T is maintained that the geometry of Euclid is best interpreted as an attempt to deduce as many important propositions as possible from the assumption that length, angle, area (and perhaps volume) are magnitudes uni- versally measurable by the methods that are actually employed in experimental physics. All his chief pro- positions (in so far as they are true) can be deduced from that assumption without any other. This view is supported, not by a detailed analysis of the Elements, but by a very summary sketch of the laws that must be true if the assumption is to be acceptable. In a sequel it is hoped to discuss similarly the foundations of another branch of experimental geometry with which BHuclid is not directly concerned—namely the geometry of position, which involves the concept of ‘‘ space.” 1. There was formerly much discussion whether geometry was an experimental or a mathematical science. It is now generally agreed that there are two closely connected sciences, one mathematical and one experimental. The former, which has been defined as the study of multi- dimensional series, consists of a logical development of ideas which have no necessary dependence on the experience of the senses. It does not consist of laws and cannot be proved or disproved by experiment; it can enter into relation with experimental science only through theories and by suggesting hypotheses which, interpreted suitably, predict laws. The formulation of such theories, in which Minkowski was the pioneer, is one of the most striking features of modern mathematical physics. The experi- mental science, on the other hand, is meaningless apart from experience, and its propositions are true or false according as they agree or disagree with experiment. They are the very fundamental laws which involve only the geometrical magnitudes such as length, angle, or area. It may be noted in passing that the laws predicted by geometrical theories are not in general geometrical laws, but involve electrical, optical, or dynamical concepts. * Communicated by the Author. 16 Dr. Norman Campbell on the The mathematicians who have recently taken over from the philosophers the task of teaching experimenters their business have decided that only the mathematical science is properly termed geometry. In support of their claim they appeal to the authority of the Greeks, and thereby imply that Greek geometry is mathematical and not experi- mental. This implication raises questions of scientific interpretation and not of mere convenience in nomenclature. For the matter cannot be dezided by inquiring what Euclid (for example) thought he was writing about: it is admitted that, as an exponent of mathematical geometry, he was guilty of errors ; and, if he was capable of error, he may have been wrong as to the nature of his assumptions and of his arguments. If we are justified today in confining the term to one study rather than another, because that term was used by Euclid, it can only be on the ground that Huclid’s propositions and his methods of proving them are closely similar to those employed today in that study. If this test is applied, geometry is an experimental science. For whereas the Hlements is utterly different from anything modern mathematical geometers produce, it is, judged by modern standards, quite a creditable attempt at an exposition of experimental geometry. It can be regarded broadly as an attempt to deduce as many important laws as possible from the single assumption that length, area, angle, and (less definitely) volume are magnitudes, universally measurable by the methods which are actually employed in experimental physics, or to which the methods that are actually employed would be referred if doubt arose concerning their validity. Nothing is assumed but that every straight line has a length, every pair of strarght lines an angle, and every plane surface an area. The definitions, axioms, and postulates should then be state- ments of the laws by virtue of which measurement is possible. It is admitted that the attempt is not wholly successful ; but its faults, or many of them, are readily explicable: the author has not to be represented (as he must be if he is an exponent of the mathematical science) as constantly straining at gnats and swallowing camels. Such a view can be established only by a detailed and tedious criticism which, in so far as it concerns Euclid’s intelligence, is not of scientific interest. In place of it will be offered a very summary sketch of the fundamental notions and laws of experimental geometry and sufticient comparison of them with Euclid’s assumptions to suggest that on them might be founded a deduction, by methods on ee Elements of Geometry. 17 very similar to those that he employs, of the propositions which he actually states. References are throughout to Todhunter’s edition. (2) But two preliminary questions must be asked. First, can an experimental science be deductive at all? Certainly it can. A deduction from a law is an application of that law in particular circumstances which were not examined when it was formulated. If, after examining the sides of squares and of triangles, I assert the general law that all straight lines have measurable lengths, and then, without further ‘experiment, assert that the diagonals of squares, which are also straight lines, are also measurable, I am making a deduction. It may be true that there is some- thing precarious about the results of such deduction— that question is not raised here,—but the deduction itself is quite unexceptionable; the falsity of the conclusion is definitely inconsistent with the truth of the premises. If deubt is raised concerning the conclusion, the ultimate means of resolving it is by experiment; but experimental science, in the hands of its greatest exponents, consists in asserting such general laws that doubt does not arise concerning the results of deduction based on them. The second question is whether there are truly laws which make measurement possible. The question is dis- cussed at length in my ‘ Physics,’ Part II., the results and nomenclature of which will be used freely in what follows. But there is one matter which may receive special mention here, beeanse it is concerned with ‘“ incommensurables,” which are often (but falsely) believed to be of especial importance in geometry. Measurement is possible when, by means of definitions of equality and addition, a standard series of the property in question can be established, starting from some arbitrary unit, such that any system having the property is equal in respect of it to some one member of the standard series. Now (it might be argued) such mea- surement is not possible for length, because the diagonal of a square cannot be equal to any member of a standard series based on the side as unit; indeed that result is actually proved by Huclid. Consequently it is patently absurd to pretend that Euclid’s propositions can be derived from an assumption, namely that measurement is possible, which is inconsistent with its conclusions. One method of escape from this difficulty may be mentioned, although it will not be adopted. A slight Phil, Beg S. 6. Vol. 44. No. 259, July 1922. ‘6: 18 Dr. Norman Campbell on the amendment in the thesis might be made, and it might be said that Euclid’s assumption is that the laws are true which would make measurement possible if there were no incommensurable lengths—for these laws, though necessary to measurement, may not be sufficient. But the diticulty vanishes entirely, if it is remembered what is meant by “equality” in experimental measurement. When it is said that A is equal to B, it is meant that there is no possible means of deciding which of the two is the greater. If then I say that the diagonal of a square is 2 times the side, I mean that, if I measure the diagonal in terms of the side as unit, there is no means of deciding whether | the value obtained, when multiplied by itself according to the multiplication table, will be greater or less than 2. That statement is not in the least inconsistent with my assigning to particular diagonals values of which the square is not 2; it is only inconsistent if a law can be found by which I can tell in particular cases whether the square will be greater or less than 2. My assertion is that. there is no such law ; and that assertion is true. In its appli- cation to all magnitudes except number, equality must be interpreted in this, slightly statistical, sense. 3. There is then no preliminary objection to the view that Euclid’s propositions are deductions from the laws in virtue of which the geometrical magnitudes are measurable. We now proceed to ask what those laws are. Geometrical conceptions are derived ultimately from our immediate sensations of muscular movement, just as dynamical conceptions are derived from our sensations of muscular exertion and thermal conception from our senge of hot and cold. We have an instinctive and indescribable appreciation of differences in dzrection of various movements ; we appreciate that one direction may be between two others ; and if other sensations (e. g. those of hot and cold or rough and smooth) vary with movement along a certain direction, we appreciate that of the varying sensations some are between others. The notions of direction and of the two kinds of betweenness are the foundations of geometry. It is a vitally important fact that there is an intimate relation connecting betweenness determined by one kind of muscular motion (e. g. that of the hand) and that determined by another (e.g. that of the eye). ‘The relation is much too complex for any account of it to be attempted here ; but it is only because it exists that “space” explored visually or by our different limbs is always the same. ee Elements of Geometry. | 19 The fundamental notions give rise to those of surfaces and lines. Surfaces are connected with the fact that a sensation may be unaltered by movement in any of a certain group of directions (which are said to be in a surface and to cha- racterize it), while it may be altered by any movement in any direction not in this group (directions away from the surface). OF lines there are two kinds, which will be termed respectively ‘edges”’ and ‘ ‘ scratches.” Hdges arise from the fact that the group of directions characteristic of a surface may change suddenly at some part of it. It is a matter of convenience whether the parts characterized by different directions are spoken of as different surfaces or as parts of the same surface: we shall adopt the second alternative. Scratches arise from the fact that, while the directions characterizing a surface are unaltered, the sen- sation the occurrence of which distinguishes “in the surface’ from ‘‘out of the surface” may change suddenly. Some, but not all lines, are such that the whole of them hes along a single direction. Points are of little importance in the earlier stages of geometry ; they arise from the fact that two lines may have a part in common. Two points, both on the same line, are termed the ends of the part of the line between those points. The recognition of surfaces and lines is the first step towards geometry. Euclid attempts to give an account of them in Defs. 1, 2, 3, 5 of Book JI., which are the least successful part of his treatise. The account given of them here is no better than Euclid’s for the purpose of conveying a notion of them to one who does not possess it already ; but since there are no such persons, the objection is not serious. But our account is better in drawing attention to the notions that are fundamentalin geometry and in not assuming familiarity with conceptions, such as length, which are necessarily subsequent. 4. Some surfaces, but not all, when subjected to muscular force undergo only such changes as can be compensated by a suitable movement of the whole body; if such a movement is made, the group of directions characterizing the surfaces is restored. In other words, such surfaces can move without alteration of form ; they provide the original and erude con- ception of a rigid body. By means of the motions of rigid bodies, it is sometimes possible to bring parts of two pre- viously distinct surfaces into contiguity, so that there is nothing between those parts. In particular, edges, or parts of edges, can often be brought into such contiguity. Scratches C2 20 Dr. Nerman Campbell on the can be brought into contiguity with edges, and, in a sense, into contiguity with other scratches ; but the criterion of contiguity in the last case is much less direct and requires methods involving something other than the simple per- ception of nothing between. The recognition of the possibility of contiguity is the second step towards geometry and leads immediately to the third, which consists in the establishment of a definite criterion for a straight line. A crude criterion is provided by direct perception : a young child knows the difference between a straight and a bent line by simply looking at them; the recognition seems to depend on ihe fact that a straight line is all in one direction and is symmetrical with regard to the unsymmetrical directions of left and right or back and front. The crude criterion is stated as well as it can be in Huclid’s Def. 4. But contiguity provides a much more stringent criterion, which in the first instance is applicable only to edges and not to scratches. Two edges are straight if, when two portions of one are brought into contiguity with two portions of the other, all the portions between these two portions are also in contiguity, however the contiguity of the first pairs of portions is. effected. It appears as an experimental fact, that if A,B and ©, D are two pairs of straight edges according to this criterion, C is also straight if tested against A; accordingly an edge can be called straight independently of the other member of the pair on which the test is carried out. A seratch is straight if it can be brought into complete contiguity with a straight edge, These facts are stated in Axiom 10. Other definitions of a straight line are sometimes offered : e.g., (1) an axis of rotation, (2) the shortest distance between two points, (3) the path of a ray of light. (1) is almost equivalent to that stated here; (2) will be noticed presently ; (2) is not accurately true (7. e., if it is adopted, the familiar propositions about straight lines are not true), but it is important as an approximation for comparatively rough measurements. A plane surface (or, according to our usage, pari of a surface) is then defined as in Def. 7. It can also be defined by the complete contiguity of three pairs of sur- faces ; but the contiguity of surfaces is not easy to describe accurately. Such a definition is, however, actually used in making optical flats and surface plates ; if it were adopted, it would still be necessary to introduce the fact that it aorees with our definition, in order to measure angle. The 79 F 2 Elements of Geometry. 21 conception of the contiguity of surfaces is not actually required, except perhaps for the measurement of volume. (Cf. § 11.) 5. The third step places us in a position to introduce measurement and the three fundamental magnitudes, length, angle, and area. For tundamental measurement we need definitions of equality and addition, such that the law of equality and the two laws of addition are true. The choice of unit may be left out of account ; for, with geometric magnitudes, the laws are true whatever unit is selected. The law of equality is Axiom 1; the first law of addition is Axiom 9. Axioms 2-7 are together very nearly equivalent to the second law of addition (which may be stated roughly in the form that the magnitude of a sum _ depends only on the magnitudes of the parts). Axiom 8 is an attempt to compress the definitions of equality for all three magnitudes into a single sentence; it is better to separate them. Huclid fails to give any definition of addition : he does not tell us how the “ whole” is to be related to the “ parts” in order that it should be greater. 6. We will now take the magnitudes in turn. For the length of a straight line the necessary definitions are :— (1) Two straight lines are equal in length if they can be placed so that when one end of the first is contiguous with one end of the second, the other ends are also contiguous. (2) The length of the straight line AB is equal to the sum of the lengths of the straight lines CD, EF, if they can be placed so that C is contiguous with A, F with B, D with E and with some part of AB between A and B. These definitions, like all similar definitions of mag- nitudes, are satisfactory and are subject to the necessary laws of equality and addition only if certain conditions are fulfilled. The conditions are described by saying that the surfaces in which the straight lines lie must be those of rigid bodies. This is a definition of a rigid body: a rigid body is something which (like a perfect balance) is determined by the satisfaction of the conditions for mea- surement*. Rigid bodies according to this test include many of those which satisfy the crude test of § 4, though they include others (e.g., surveyors’ tapes used as surveyors use them) which do not satisfy that test. In virtue of the fact that rigid bodies are necessary to measurement, the * Cf. H. Dingler, Phys. Zeit, xxi. p. 487 (1920). 22 Dr. Norman Campbell on the branch of geometry with which we (and, according to our view, Huclid) are concerned may be fitly deseribed as the study of the surfaces of rigid bodies. It is thus dis- tinguished from a wholly different branch of geometry, with which we are not here concerned, ihat is not confined to rigid bodies ; this is the geometry of position. It is important to notice that not all pairs of straight lines can be brought into contiguity, and that the law of equality cannot therefore be tested universally. It might have turned out that there was some material difference between those which can and those which cannot be brought into contiguity with a given line; and that if we assumed that the law of equality is universally true, we should be led to inconsistencies. It is an experimental fact that no such inconsistencies do arise when we extend our definition of equality so that lengths are equal when they are equal to to the same length, although they cannot be brought into contiguity with each other. This is, of course, one of the most important laws that make measurement possible. A similar remark applies to all the geometric magnitudes and need not be repeated. 7. The length of lines that are not straight can be measured approximately as fundamental magnitudes by means of flexible but inextensible strings. But the laws of such measurement are not strictly true, because (as we say now) no string is infinitely thin and the surface never coincides with the neutral axis. Another possible way, perhaps more accurate but of limited application, would be to roll curved edges on some standard edge, which ~ need not be straight. But in truth there is no perfectly satisfactory way of measuring fundamentally the length of curved lines. All the measurements which we make on them are derived from measurement of straight lines ; they involve numerical laws between fundamentally measured magnitudes. One of these laws is that the perimeters of the circumscribed and inscribed regular polygons tend to a common limit as the number of sides is increased. That law is therefore a law of measurement if curved lines are to be measured. The question whether curved lines can be measured fundamentally is important, because, if they could be, it would be possible to define a straight line as the shortest distance between two points. (The definition would have to be put in some other form, since distance, a conception belonging to the geometry of position, implies the mea- surement of length.) But since they cannot be, that Elements of Geometry. 23 definition must be rejected ; it must be regarded merely as a generalized form of Prop. I. 20. 8. Angle is the measure of the crude conception of direction. The following are the definitions of equality and addition for the angle “between two intersecting straight lines :—The angle between two straight lines A, B is equal to that between C, D if it is possible to bring "A into con- tinguity with C and B with D. The angle between A, B is the sum of the angles between CO, D and | Dp elistr oe ‘when A is brought into contiguity with C and D with E, D lying between C and F and in the same plane with them, F can be brought into contiguity with B. These definitions are satisfactory only if the straight lines are in rigid bodies; or, in other words, there are surfaces which satisfy fhe conditions for the measurement of length and also those for the measurement of angle. But even if the surfaces are those of rigid bodies, the definitions are not wholly satisfactory and the laws of measurement not entirely true. We must distinguish angles according as the two straight lines which they relate are or are not prolonged on both sides of the common point: the latter class may be termed “ corners,” the former “‘ crossings.” Angles between edges are always corners; those between scratches may be either corners or crossings. If we try to include both corners and crossings in the same class as a single magnitude, the law of equality is not true; for two corners which are both, according to the definition, equal to a crossing may not be equal to each other; as we say now, one angle may be the supplement of the other. But if we treat corners and crossings as separate magnitudes this difficulty disappears ; the law of equality is true for either taken apart from the other. Actually we take corners only as magnitudes ; crossings we measure by the corners with which they can be made contiguous. Hach crossing then has four angles (2. e. corners) associated with it. It is an important experi- mental fact that the “opposite” angles are equal; it is best taken as a primary law, instead of being proved from other axioms asin Prop.I.15. It is a law of measurement, because if it were not known, we should need four and not two angles to measure a crossing ; it is thus inherent in our system of measurement. But though the law of equality is now true, the first law of addition is ‘false ; ; it is false for both corners and crossings. The whole which is the sum of the parts may be equal to 24 Dr. Norman Campbell on the one of the parts: e.g., if both of two parts, being corners, are what we now call 120°. Some kind of spiral space can be imagined in which the law would be true; but actually it is very important that it is false. For, apparently in- separable from its falsity is the fact that the angle between two portions of the same straight line can be measured and given a finite value in terms of a unit which is the angle between two intersecting lines. This fact is described by the assertion that there are right angles and that a per- pendicular can be drawn to any straight line from any point in it, a right angle being defined as in Def. 1. 10. (Axiom 1. 11 follows from this definition, regarded as an existence theorem, and our axiom Prop. I. 15.) Since the existence of right angles is vital to geometry, we cannot avoid the falsity of the first law of equality by some alteration of the definition. We can only recognize that the law is true in some conditions, and be careful to apply it in deduction only when it is true. It is true when all the lines making the added angles lie on the same side of (or contiguous with) a single straight line passing through their commen point; this condition can be expressed, though with some complexity, in terms of the fundamental notion of between. Thus, in proving Prop. I. 16 we need to know that OF and CD both lie on the same side of AG. . This law, and perhaps others of the same nature, are laws of measurement, defining the conditions in which angle can be measured uniquely. They require explicit mention. The ambiguity which the falsity of the first law of addition introduces into numerical measurement is removed by certain conventions. These need not be considered here for we are not assigning numerical values. If the length of curved lines were measurable funda- mentally, angle might be measured as a pure derived magnitude, e.g. by the ratio of the are to the radius of a circle in virtue of the numerical law, established experi- mentally, that the are is proportional to the radius. But since curved lines cannot be so measured, we must take angle to be fundamental. We cannot use right-angled triangles with straight sides to measure angle as derived, because we need fundamental measurement to determine what angles are right. Of course we might define for this purpose a right angle as an angle between some two lines arbitrarily chosen as standard ; but such measurement would be intolerably artificial and nothing whatever could be deduced from such a definition. . pie ee e Elements of Geometry. 25 9. Euclid’s definition of parallel lines must be rejected entirely, for, since all plane surfaces are limited, the criterion suggested is inapplicable. Since the crude de- finition of parallelism is similarity of direction, we may try to define parallel lines as those which being in the same plane make the same angle with any third line. We thereby imply the axiom of parallels in the form (Prop. I. 29) that such lines which make the same angle with one straight line make the same angle with any other ; we imply also that the angles which are to be equal are the “exterior” and “interior” opposites or the ‘alternate ” angles, since if the interior angles are compared the proposition is not true. But the definition is not very satisfactory ; for, when the lines are edges, there is not always an exterior or an alternate angle. It is better to adopt the substance of Axiom [. 12 as a definition, and to say that lines in one plane are parallel when the sum of the interior angles is equal to two right angles. This much abused axiom seems to me a very ingenious way out of a real difficulty. We then assert the axiom of parallels in the form (implied by I. 32) that if any two straight lines in a plane are cut by any third line, the sum of the interior angles is the same for all third lines. The merit of this axiom is that it indicates clearly that the ‘“‘axiom of parallels” is really something concerning all straight lines in a plane and not only parallel lines, and that parallel lines are merely a particular case of other pairs of lines. The propositicns that parallel lines never do intersect and that the angle between them is zero follow immediately. The axiom of parallels is a law of measurement because it is involved in the measurement of the angle between lines which do not intersect. Its use for this purpose requires that at some point of a straight line it should always be possible to place a straight line parallel to a given straight line. This proposition is not true for concave surfaces, but the complexities arising from this failure and the means of avoiding them may be left for the present ; they are dealt with more naturally in connexion with “space.” If the axiom were not used, we could not by our present methods measure the angle between non-intersecting straight lines : first, because the definition of equality given above, though sufficient for such lines, is not necessary : second, because the definition of addition is wholly unsatisfactory. There has been so much discussion of the necessity of the 26 Dr. Norman Campbell on the axiom of parallels that the matter requires rather more con- sideration. Two questions are involved. First, would it be possible to measure the angle between non-intersecting lines without assuming some proposition logically deducible from the axiom? It would be if, and. only if, some property, common to all lines between which the angle is the same, can be found which is determinable by direct experiment not involving parallel lines. There may be such a property, but I have not been able to think of it. Second, if the axiom were not actually true—but we may stop there. In a pure experimental science, there is no sense in asking what would happen if the world were other than it actually is. Theory is necessary to give such a question a meaning, by suggesting what might remain unaltered during the change. For our present purpose the axiom is as necessary as any other of those we are considering. 10. Area is distinguished from all other fundamental magnitudes because the definitions of equality and addition are inseparable. They may be expressed thus. The areas of two bounded plane surfaces are equal if (but not only if) their boundaries can be brought into complete contiguity with each other or with the same third boundary. (A bounded surface is a part of a surface which includes all portions which can be traversed without crossing the boundary line.) The area of A is the sum of the areas of B and OC, if when parts of the boundaries of B and C are brought into contiguity with each other, the remaining parts of the boundary can be brought into contiguity with the boundary of A. In virtue of the fact that parts of the boundaries of two surfaces can be brought into contiguity in many different ways, there may be many different bounded surfaces, of which the boundaries cannot be made contiguous, which are the sum of the same bounded sur- faces. If the measurement of area is to be satisfactory, these surfaces must also be deemed to have equal area, and the definition of equality must be extended correspondingly. With this extension the laws of equality and addition are true, and the measurement is satisfactory. In order that all bounded plane surfaces should have areas, some rule must be found for choosing the shape of the members of the standard series and for grouping them in such a way that some sum of them is equal to any area. We use for this purpose rules based on the axiom of parallels, and that axiom is therefore again a law of the . ) Elements of Geometry. 2T measurement of area. ‘The rule might possibly be dis- pensed with, if we were prepared to spend unlimited time in selecting by trial and error shapes for the members of the standard series which fulfil the necessary conditions ; but actually we could never measure area except by making use of similar figures, the production and properties of which depend wholly on the axiom of parallels. Further, it is the use of that axiom which enables us nowadays to calculate area from the linear dimensions of a surface without resorting at all to fundamental measurement. But of course all the numerical laws on which that cal- culation depends have to be established by means of fundamental measurement. It is only by defining area as we have done, and assuming the axiom of parallels, that we can prove by deduction that the area of a rect- angle is proportional to the product of its sides, or equal if the units are suitably chosen. The areas of surfaces that are not plane cannot be mea- sured fundamentally, even to the extent that the length of curved lines can be. For there are no inextensible surfaces which can be brought into contiguity with surfaces of any curvature. Weasurement of curved area is always derived and estimated by the limit of the circumscribed polyhedra as the number of their sides is increased. But the whole matter is obscure, because it is much more difficult to establish experimentally that there is a limit or to say what the limit is; for there is here no inscribed polyhedron tending to the same limit. There is singularly little experi- mental evidence for the assertion that the area of a sphere is 4777, and there is great difficulty in saying exactly what we mean by such an assertion ; curved area is almost always a hypothetical idea and not an experimental magnitude at all. 11. Volume is a property of complete surfaces. Since complete surfaces can never be brought into complete con- tiguity, volume cannot be measured fundamentally by any process at all similar to those applicable to the magnitudes we have considered so far. Volume is measured (a) as a fundamental magnitude by means of incompressible fluids, or (2) as a derived magnitude by means of the lengths and Silos characteristic of the surface. The second method depends upon numerical Jaws established by means of the first. In certain cases these laws can be related closely to other geometric laws by means of the following propo- sitions :—(1) Two complete surfaces with equal dimensions, 28 Dr. Norman Campbell on the 2. e. with equal lengths and equal angles between them, have equal volumes. (2) If two complete surfaces have each one ae plane, and the boundary of the plane part of one can be brought into complete contiguity with the plane part of ie other, then the complete surface which has dimensions equal to that of the complete surface so formed has a volume equal to the sum of the volumes of the original surfaces. These propositions could be used as definitions of equality and addition in a system of measurement, which would be independent of the measurement of length and angle (and therefore not derived), because it sales only equality, and not addition, of length and angle. But it is of limited scope and, in particular, “would not permit the measurement of the volumes o£ curved surfaces. Since we do undoubtedly attribute a meaning to the volume of such surfaces, in a way that we do not to their area, measurement by incom- pressible fluids, which is not geometric, cannot, be wholly avoided. But the propositions, which are those on which — BHuclid bases his treatment of volume, are actually used in modern practice, and are therefore regarded per missibly as laws of measurement. 12. In deducing Euclid’s propositions from the laws of measurement of these magnitudes, subsidiary laws are required, corresponding roughly to his postulates, expressed and implied. TWirst, we need ‘existence theorems” corre- sponding to each of the definitions; for example, the definition of a plane surface justifies the conclusion that a straight edge ean be placed contiguously to any two portions of such a part of a surface. Second, we need the assumption that we can make an object having a magnitude equal to that of any object presented to our notice. All these propositions are laws of measurement : the first group, because ail definitions in experimental science are nothing but existence theorems; the second, because it is implied in the fact that we can make a standard series by which we can measure any magnitude. Euclid’s three expressed postulates are all untrue. I cannot ‘“ draw a straight line” from this room to the next when the door is closed. Moreover his constructional propositions, closely connected with the postulates, are unsatisfactory because they are all directed to the drawing of scratches, rather than to the making of edges. The hypothetical experiments by means of which the deductions are effected are carried out much more easily with edges Elements of Geometry. 29 than with scratches ; and if any of the propositions were donbted and put to the test of experiment, it would certainly be by means of edges ; the extension to scratches would be by means of the contiguity of edges with them. Huclid’s methods here undoubtedly indicate that he is leaving, perhaps consciously, the realities of experimental science for the pure ideas of mathematics. But he has made so little progress towards the new peak that, if he is to be restored to safety, it is far easier to drag him back to that which he has never left completely than to guide him through the bog in which the two sciences are confused to the very distant goal. 13. Only a few disconnected remarks will be offered here on the process of deducing the Huclidean propositions from the fundamental laws that have been sketched. Of course, we should employ the “application” (or contiguity) method of Prop. I. 4 wherever possible, instead of trying to avoid it; for it is based directly on the fundamental notions. Again, we should not commit Huclid’s error of supposing that strictly similar triangles can be brought into contiguity; we should apply the mirror image first to one triangle and then to the other. There would be no need to introduce area to prove Prop. 1.47. A Greek writer was forced to do so, because, not being familiar with the multiplication table, he could describe in no other way the relation between a number and its product by ‘itself. We should proceed from Prop. I. 34 to Book VI. and prove Prop. I. 47 by drawing the perpendicular from the right angle to the hypoteneuse and using the relations of similar triangles, treated by algebra. For nowadays, since we admit no incommensurable magnitudes, we can dispense altogether with Huclid’s very beautiful and ingenious subtleties about ratios. A ratio in experimental science is nothing but a value taken from the multiplication table, which is established by the measurement of number, i. e. by counting. The laws of the measurement of number are involved in those of the measurement of every ‘‘continuous” magnitude. April 22, 1922, Ill. On the Rotation of Slightly Elastic Bodies. By DorotHy WrincH, D.Se., Fellow of (Girton College, Cambridge, and Member of Research Staff, University College, London™. HE change in dimensions of a slightly elastic body due to rotation is a question of some practical importance, and does not appear to have received any systematic treat- ment. In the theory of elasticity, the displacements of a point of the body are of course discussed and the displace- ments of the points of the boundary determine the increase of dimensions. But the problems of elasticity which are of interest mainly from the point of view of increase of dimen- sions, rather than of the distribution of stress in the material, can rarely be solved by the current methods or appear only as special cases of a general mode of analysis. ven the simple problem ofa circular cylinder of finite length, rotating about its axis, has not yet admitted an exact solution, though an approximate solution, which becomes valid when the cylinder is of infinite length, has been given by Chree. When the cylinder has a finite length, the surface con- dition of zero traction over the curved surface is violated, and instead of this traction becoming zero at all points on the surface, only its average value over the surface is zero. The results for the case of an infinite cylindrical annulas do not appear to be on record, and they are interesting on account of their marked divergence from those which belong to the complete disk. In the present paper we group together some of the simpler and more interesting solutions of problems of this type, including those of the infinite circular cylinder and the infinite cylindrical annulus. These specific pro- blems are solved to any degree of approximation and for a non-uniform distribution of density. The analysis is simpler than is usual, for it does not seem necessary to treat these comparatively simple problems as special cases of general theory, and it is desirable, at least in the interests of the engineer or physicist, that a fundamentally simpler treat- ment should be placed on record. It also seems possible that such solutions may be of interest with regard to scientific instruments of great precision, in which some portion of the apparatus is in rotation, or, on the larger scale, in problems of practical engineering. Although no * Communicated by the Author. — On the Rotation of Slightly Elastic Bodies. ol novelty attaches to some of the earlier results, it seems desirable to include them. The simplest problem of this nature is, of course, that of the thin circular hoop rotating about its centre. When such a hoop of radius a and density p is spun round its centre with constant angular velocity w the value of T, the tension per unit length in the hoop, is well known. Tor an element ds of its length has an acceleration aw? inwards, and the resultant of the tensions at its endsis Tds/a per unit area inwards. Hence the equation of motion is Tds/a=paow’, me ra et giving T= pa’. If, however, the hoop is slightly elastic, and \ the value of Young’s modulus for the material of which the hoop is made, and v the radius of the hoop when in motion, the equation of motion of the stretched element ds becomes T/r=re? . pa/r. Applying Hooke’s law to the stretched element, we have, T=A(r—a)/a. Hence eliminating T, praw?=r(7r—a)/a. In practice X is always large, and if we may neglect 1/X and higher powers of 1/X the appropriate value of 7/a, which differs from unity by a quantity of order 1/d, is 1+ pa?w?/X. The value of the tension to the same order is pa’w’. The effect of a rotation is therefore to increase the radius a of the hoop to a(1+ ), where ~=pa?w?/A, a number depending on the density, the elasticity, and the radius of the hoop, and on the rate at which it is rotating. As regards the practical order or magnitude of pa?w?/r the extension per unit length, we may take a steel wire for which X is about 2°12x 10” dynes per square centimetre, and p isabout 7°5. In order that Hooke’s law may hold, the extension per unit length must not exceed 107%, roughly speaking. If the velocity of a point on the rim is in the neighbourhood of 1:°9x10* cm. per second—which is ap- proximately the case in a twenty-foot flywheel making two hundred and fifty revolutions a minute—we find that the extension per unit length is about 7°9x*107*, which comes within the limits of applicability of Hooke’s law, and that 32 Dr. Dorothy Wrinch on the the actual increase in the radius is about a fifth of an inch. In this case the tension is about 1°6 x 10°. It is further evident that / nd/p is the largest velocity if an extension of more than n per unit length is to be avoided. When the elastic limit for the material is known, this result can be used to give an upper limit to the velocity it is safe to use if risk of deformation of the hoop is to be avoided. | F We may now proceed to the problem of a thin rod rotating about one end with uniform angular velocity. Thin Rod Rotating about One End. Let ay be the unstretched length of the rod, @ the angular velocity of rotation about one end QO, po the density when it is unstretched, and » the value of Young’s modulus for the material of which the rod is made. Let T be the tension in any section in the rod during the motion. Let the distances of the same particle at rest and in motion be wy, and w The density of the moving element dw is poda,/dx and its acceleration towards O is ww’. The equa- tion of motion of the element is therefore, OT = — py bay. 2o’, where, by applying Hooke’s law to the element originally of length da) and now of length dz, we have T=A(dx/da—1). Hence, eliminating T, we obtain the equation, dx /day? = — pow] d. The solution must give the value «=a when #=a) if a is the length of the rod when in motion. Accordingly it is a=asin (& Vpyw?/d) / sin (ay ¥ pow?/2). We may determine the value of a by means of the condition that the tension vanishes at the free end, which is given indifferently by e=a or aj=ay. Thus, QV pyw?/A=tan dg V pow?/d. The equation relating the two corresponding positions of a Rotation of Slightly Elastic Bodies. 33 typical element when at rest and when in motion and the original length of the rod is therefore ® v/pyw?/A=sin (xy Vpow?/A) / cos (ao WV pow?/X). Neglecting the cube and higher powers of 1/A, we may replace this by the simpler form, L=Xq + Lppow? (3a? — xp”) /6X to the order 1/A. To the same order, T= po@?a,? [| 1—2y?/a)" |. The greatest extension is $pow’a,?/d, and this occurs at the end about which the bar is rotating. The tension is also greatest at this point and takes there the value p)wa,”. As an example of the actual magnitudes of the quantities in practical cases we may take a twenty feet steel bar, which, when rotating about one end two hundred and fifty or three hundred times a minute, increases in length about a tenth of. an inch. Rotation of an Infimte Elastic Circular Cylinder about its Aas. Passing now to a simple problem in three dimensions, we take the case of an infinite elastic cylinder of circular section rotating about its axis. We may consider one of the circular sections of the cylinder and use polar coordinates. At any point (7, @) let I, and T, be the transverse and radial ten- sions per unit length, and T; the axiai tension. We shall consider the motion of the element of volume which when at rest is bounded by the surfaces (z,<2+6z), (r, r+é6r), (0,0+60@). By the symmetry of the cylinder, the element when in motion will continue to be bounded by the surfaces (0,9+60): and since the cylinder is of infinite length, the element will continue to be bounded by the surfaces (c,¢+6:). Let p represent the radial dimension, so that —r is the radial extension at any point. Let a be radius of the cylinder and o its density, when at rest ; let w be the angular velocity of the cylinder about its axis, and A and yw the elastic constants for the material of which the cylinder is made. The element of volume which we are considering is a parallelepiped of sides dp, pd@, and dz. The forces on our element of volume consist of (1) transverse tensions each Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. D 34 Dr. Dorothy Wrinch on the of magnitude T,dpdz—and these are equivalent, in the usual way, to a radial force towards the centre of magnitude T, dp dz ds/p, where ds=pd0; (2) of radial tensions T,dsdz and T,dsdz+dpd(T. ds dz)/dp, towards and away from the centre, which together give a oe dp d@ dz d(pT.)/dp away from the centre ; and (3) of longitudinal tensions each Fig. 1. 7, Sp. 6p Ee (7p ds) of magnitude T;dp ds, in opposite directions. The resultant force then is simply dp dz ds (d(pT,)/dr —T;) away from the axis, and perpendicular to the axis of the cylinder. The acceleration of the element is pw’ towards the centre and its mass is ordrd@, since we may, of course, treat the density as constant over the element of volume. We therefore have the equation of motion, d(pT,)/dp-- T,;= — orp? dr/dp. Rotation of Slightly Elastic Bodies. 30 By Young’s law we can express T, T., and T; in terms of the extensions of the cylinder, in the well-known equations, =X(dp/dr—1)+ (A+ 2)(p/r—1), = (A+ 2u)(dp/dr—1)+A(p/r—1), Ts; =A(dp/dr + p/r—2). Putting these values of T, and T, in the above equation we obtain the result d dp p SEP (pea, eae Buea dy [p(X +n) (S 1) + pr", 1) | AS Oe Ze -1)- 2 O23) (0-1) =—orpo dr or (pir) dpidr? + (Lir) [(2u dpfaryi-+ 2) + pir) (4 2u)] | dp/dr—p/r|=—pow?/(A+2y). . (1) The value of ww or A varies from about 8 x 108 grammes’ weight per square centimetre for steel to about 4x 10° grammes’ weight for copper. The corresponding densities are about 7 and 9 respectively. ‘Terms involving o/(A+ 2) are therefore of a smaller order than those which involve coefiicients of the form 2u/(A+ 2u). Let us write @’o =q(A+2yu). Then putting p—r= nt eno ans + we can obtain a value for p to any order of approximation which is required. Neglecting, first, all terms involving 9’, we have the equations for nN; p=r+qm; dpfdr=1+qdnifdr; ad’p/dr?=q d’n,|dr’ ; g (149 m/r] dm/dr? +1 ]r [1 +9 2p dm/adr)/(\ + 2p) + (An,/r)/(A+ 2p) +9 (dmi/dr—m/r)\=—gr(1t+q mr) ; q d’n,[dr? (1 +4 m/r)+1/r (1+q Qp dn, dr)/(X + 2p) + (Am/1)/(A+ 24) ] 9 (dm/dr—,/r) + gr [1+ 9m/r] =0; and since we are neglecting terms in gq’, the equation for 7, is re y d?x,/dr? a+ Us dn,/dr—n, => D2 36 Dr. Dorothy Wrinch on the giving a solution of the form m= ay + b,/r—$r? and 6;=0, since m, is not infinite at the axis. The boundary ~ conditions determine the constants a, and b,; for the radial traction T is given by : d T= (A+ 2u) (* -1) +r(2-1) =¢ | (A+ 2u)dy,/dr +r 9,/7], and the radial traction must vanish over the two boundary surfaces. Thus (A+ 2p) dnyfdr | pea +X m/7| pe =0. The constant a, is therefore determined by the equation, (A+ w)a, = (204 34) a?/8, giving Ti=¢ [2+ p)ay—7?/4(204 34) | =9(2r == dm)(a?— r) [40° and m=q(ar—r'/8) = [20+ 3p)(@+0)/A+p)—77], The Lffect of a Circular Hole in the Cylinder. If the cylinder at rest has two boundary surfaces r=a, r=b (b 40 Dr. Dorothy Wrinch on the Our second approximation is therefore, p—r= qn + Fne ge vietie 2u+3y ae agr[ Rta 9) y g +g] 2N+ 3M 5 a(- Ae 18) N+ ph N+ tea al cans Any higher degree of approximation can also be obtained. Our results stand in the form, 1/7 = 9% + 14,7”, No|T = ola + 1Ag?” + oo, and, in general, the form will be hel Plea. 2 lt ar) eee ei) We easily see that if (2) be the form for 7/7, then’ Pc Aah me at jar" Suga a een tny bbe eee isn PEG aaa |! and the condition of zero radial traction, requiring that (N+ 2) dnsai/dr+rqo41/7 =0 on r=a, yields ots Zr+3pm , 9s 3A + Db A Wels er Kb Neer ED ss (s+1) A Zsa a, 25+2.2s+4 N+ pe so that k ls (2R4 3m ods (ON+9M 4 4) Ns+i/T= ral eee nant aes a ae ss (s+ IN+2s+1p qzst2 — oo) a Ss Sw i Oo Pesug. Isr a\ N+ pb a 2) When we have obtained a solution, p—r= 9m tq'ne+. + +9°Ns, it is therefore possible to obtain a solution to a higher degree Rotation of Slightly Elastic Bodies. 41 of approximation by remarking the relation between the co- efficients in 9/7 and y,4;/7 and putting instead of s adi Linde ? "5 0 as In 9/7, Ns 41/7 = $ (“ e 1)r a (2n +4 Me gina — pons) oe: not aes’, n=0 N+ ph (Zn -- 2)(2n+ 4) Rotation of an Infinite Circular Cylinder of Non-uniform Density. We may next deal with the case of a cylinder in which the density is a function of the distance from the axis. Treating the case of the solid cylinder, we may put for the density of the cylinder when at rest, N c=f(r)= Da”, 0 and N may have any value from zero (when the density is uniform) to infinity—in which case the series Ya,7” must be convergent. | The equation to be solved for a first approximation to the value of 7 is as before, dy[dr? + 1/r dn[dr—y/7? = —ow'r] (A+ 24). The solution is evidently, aon is n=Ar— eae > ann’ **i((n+3)?—1). The fact that there must be zero radial traction on the boundary surface r=a, yields the condition, T,= (A+ 2p) dyn/dr+rn/r=0 on r=a, giving 2(A+p)A 2 N iF cone [(n43)(X+2u) +2] a,a"*?/(n+4) (n +2) 2 N . + ieee ee [(n+4)X+42(n+3)p] a,0"*?/(n+4)(n +2). od 0 i 42 Mr. G. H. Henderson on the Consequently, the value of T is Le S : < (Oa Le paar 9 Kb Dye L$ 3) (A+ 2p) +A] (aro) ay] +4) (m4 2), and ee p N+ N a ‘ La,fa"t?(nt4r+2n+3 p)/2(X+ wp) —7*?} /(n +2) (n +4). . . Higher approximations to the value of 7 can be obtained by the method adopted in the case of uniform density. IV. The Straggling of « Particles by Matter. By G. H. Hurnperson, M.A., 1851 Exhibition Scholar of Dalhousie University, Halifax, N.S.* . SL. Introductory. 7 HEN a parallel beam of & rays passes through matter, the particles gradually expend their energy in passing through the atoms of the matter, until all trace of the particles suddenly seems to vanish at the end of their range. In passing through the atoms some of the « particles © lose more energy than others, so that at any point along their path some of the particles will be moving more slowly than others ; also their ranges will not all be the same. The a particles may be said to be straggled out, and hence the term straggling has been applied to this phenomenon by Darwin. | The theory of the passage of matter by « rays has been developed on the basis of the nuclear structure of the atoms of the matter, and from this theory the amount of straggling to be expected has been deduced from probability. considera- tions. On the other hand, the straggling can be determined from experimental data in two ways. The first method makes use of ionization data. When the ionization due to a parallel beam of @ rays is measured at different points along the path of the rays, the well-known ionization curve is obtained. This curve is shown as the * Communicated by Prof. Sir E. Rutherford, F.R.S. Straggling of « Particles by Matter. A’ full curve of fig. 1, where ionization is plotted as ordinate and distance from the radioactive source as ubscissa. Now it has been shown experimentally that the (average) velocity of the @ particles at any point of their path is proportional to the cube root of their remaining range. Assuming that the ionization produced is proportional to the energy lost by the « particle at any point of its path, it can at once be shown that the ionization should be inversely proportional to the cube root of the remaining range. Such a theoretical ionization curve is shown as the dotted curve of fig. 1. It Rios 1 { 2 5 cm. Range will be seen to bein approximate agreement with experiment over the first portion of the path of the @ particle, but as the maximum is approached this agreement fails. Geiger * has suggested that the ionization curve observed for a beam of « rays should be different from that of a single « particle, owing to slight variations in the ranges of the latter, 7. e. to straggling. -The ionization curve, built up of a large number of theoretical curves grouped around one of average range, will thus be modified considerably near the maximum where the ionization is changing rapidly Hence the shape of the ionization curve near the end of the range should give an indication of the amount of straggling. * Geiger, Proc. Roy. Soc. A. Ixxxiii. p. 505 (1910). 44. Mr. G. H. Henderson on the Secondly, a more direct measure of the amount of straggling can be determined by counting the number of a particles at different points along the path of a parallel beam. It is proposed in this paper to discuss the theoretical and — experimental data on straggling, and it will be shown that the observed amount of straggling is much in excess of that allowed by theory. [Further experimental evidence bearing on straggling will also be brought forward. $2. Lhe Straggling in Air. It might be thought that the individual @ particles are emitted with slightly different velocities, thus giving rise to straggling. It has been shown by Geiger ( (loc. cit.), however, that the « particles emitted from a thin layer of radioactive material do not differ by as much as 4 per cent. in initial velocity. Thus the cause of the strageling must be looked for in the air itself. As the « particle passes through the air it gives up its energy to the electrons and nuclei of the air atoms, and it is occasionally deflected through a considerable angle by close encounters with the nuclei. Different a particles will encounter different numbers and distributions of electrons and nuclei and accordingly are straggled out. The calcu- lation of the consequent probability variations in the ranges of the individual « particles has been carried out by both Bohr* and Flammf. They agree in showing that the nuclei produce practically no stragoling. They. also agree closely in the amount of straggling produced by the electrons. The straggling of various types Oe rays in alr, calculated by Flamm’s method, is shown in the second ‘column. of Table, 5 Vie ln. tabulated is the distance, measured along the range, over which the number of particles in a parallel beam falls off from +92 to :08 of the original number. This corresponds approximately to the method of measuring the straggling from the experimental curves. The ionization curves for three types of a rays have recently been determined with some accuracy by the writer {. The full curve given in fig. 1 is a reproduction of the ionization curve found for RaC. It was shown that the ionization curve from C to B (fig. 1) could be represented very approximately by a straight line. The slope of this straight line furnishes infor mation as to the magnitude of * Bohr, Phil. Mag. xxx. p. 581 (1915). ay Flamm, Wien. Ber. IL a, exxii. p. 1893 (1918). t Henderson, Phil. Mag. xlii. p. 588 (1921). Straggling of « Particles by Matter. 45 the straggling. The. easiest way of considering the matter is to imagine the straight line produced in both directions till it meets the axes of zero and maximum ionization at D and H. Then the projection of the line DE on the axis of zero ionization (or the reciprocal of its slope) is a direct measure of the straggling. The greater the straggling the greater will be the projection referred to, and as a first approximation the projection may be taken as proportional to the straggling. The values of the projections taken from the writer’s curves are given in the third column of Table I. That for polonium has been determined from the curves given by Lawson *. The curves obtained by counting the number of « particles in a parallel beam at various points along the path show that this number remains constant till near the end of the range and then falls off rapidly to zero. Most of this falling off nist approximately followsa straight line. The reciprocal slope or projection of this line is a more direct measure of ' the straggling than the corresponding projection of the ionization curve. Measurement of the str ageling by means of counting experiments is, however, very slow, as large numbers of @ particles must be counted. The values obtained from the most recent and reliable counting experi- ments are given in the fourth column of Table I. The result for polonium is taken from a scintillation curve given by Rothensteiner + ; that for RaC is from a curve obtained by Makower f by photog raphic counting of the « particles. All the results in the Table are in millimetres and refer to air at 0° C. and 760 mm. TasceE I. | Gas ae. Theoretical | Straggling from | Straggling from eee oe Baye Straggling. —_Lonization ‘Curves. Counting Expts. | Polontum: .3...:... 88 31 | 40 | Thorium C, ...... 1:02 2°88 | | Radium C......... 1-44 | 2-83 | 41 2-92 | | | Thorium C,....... 1-74 This Table shows clearly that the observed values of the * Lawson, Wien. Ber. il a. exxiv. p. 637 (1915). + Rothensteiner, Wien. Ber. IL a. exxv. p. 1237 (1915). t Makower, Phil. Mag. xxxii. p. 222 (1916). AG We a Siloceénton oho straggling are three or four times greater than those predicted by theory. Furthermore the calculated straggling increases steadily with increase of range, while that observed is constant within the limit of error. It should be pointed out that the projections given in the Table are measured as the small differences between two larger quantities, and hence are more difficult to determine withaccuracy. The straggling deduced from the writer’s experiments has a probable error of about 2 per cent., and it will be seen that the values for the three types of rays agree within this limit. It was shown by the writer (loc. cit.) that the effect of straggling due to electronic encounters would be a tailing off of the ionization curve at the extreme end of the range. Making some simple assumptions it was shown that the calculated form of the end of the ionization curve agreed satisfactorily with the form of the curve observed between Aand B (fig.1). Thus the effect of the calculated straggling was amply accounted for by AB, leaving the much greater straggling evidenced by the straight line portion BC quite unexplained. ‘The curves obtained by counting experiments also lead to precisely the same conclusion. In view of the failure of theory to account for this large excess strage gling it 1s interesting to see what further information regarding it can be derived from experiment. It is remarkable that the straggling (as measured by the projections of the ionization and also of the counting curves) should be constant for arays differing so widely in range as those givenin Table I. This can only mean that the excess strag ogling takes place only in the last two or three centi- metres of the range. From experiments with gold foils which will be discussed later, it appears probable that the straggling is confined to the last few millimetres of the range. Referring once more to fig. 1, it could not be expected that the strageling deduced from the ionization and the counting data would agree, for the following reasons :—The ionization curve is revarded as being built up of simple curves of different ranges grouped about a common mean. The form of the simple curve is not accurately known ; the rule that the ionization is inversely proportional to the cube root of the remaining range can only be an approximation to a much more complicated law. As the shape of the simple curve cannot be taken into account, the projection of the ionization curve which is actually utilized can only give a Straggling of « Particles by Matter. 47 rough indication of the absolute magnitude of the straggling. However, comparative values of the straggling under different conditions should be given fairly accurately by the method adopted. On the other hand, in the counting experiments the assumption is made that the zine sulphide screens or photographie plates used have the same efficiency for « rays of low speeds as for those of high speed. This assumption is not altogether justifiable. § 3. Straggling in Gases other than Avr. The ionization curve in hydrogen was determined with the same apparatus already used for air. The gas was obtained from a cylinder of compressed hydrogen stated by the makers to be of more than 98 per cent. purity. Small impurities are unlikely to affect the straggling materially. It was again found that a considerable portion of the end of the ionization curve could be represented by a straight line. When the range of the a particles was reduced so as to give the same range as in air, the projection of the straight line was 2'05 mm. with a probable error of 3 per cent. The straggling in air and hydrogen may be deduced from the ionization curves given by other observers. ‘lhe results agree in every case within the limits of error, although the conditions for accuracy were less favourable than in the present experiments. The value 2°0 mm. for polonium in hydrogen may be obtained from some ionization curves given by Taylor*. From the results of Lawson (loc. cit.) for polonium the straggling was determined as 3°1 mm. in air and 2:2 mm. in hydrogen. The straggling in oxygen has also been deduced from experiments made in that gas with the present apparatus, using ThC. The value found was 3°36 mm., when the range was increased to the same value as in air. The straggling in several other gases may be deduced from the ionization curves given by Taylor (oc. cit.), although the error involved is probably of the order of 10 per cent. The collected results of straggling in gases are given in fig. 2, which shows the straggling plotted against molecular stopping power. The values plotted for air, hydrogen, and oxygen are from the writer’s results; the remainder are taken from Taylor’s curves. * Taylor, Phil. Mag. xxi. p. 571 (1911). 48 Mr. G. H. Henderson on the Fig. 2 can only be considered to give an approximate idea of the facts, as the points are not well distributed and some may be seriously in error. It would seem, however, that the straggling increases very slowly as the stopping power of the gas is increased. It is unfortunate that the dearth of suitable gases of high stopping power makes the checking of this point difficult. Fig, 2. Straqgling inmms. 3 1 2 Molecular Stopping Power. §4. The Straggling due to Solids. The great difficulty which at once arises in determining the straggling due to solids is the uneven thickness of the solid foils used, the effect of which may completely mask | the true straggling looked for. An attempt to avoid this difficulty was made by using a large number of the thinnest beaten foils of the solid obtainable ; with gold, for example, as many as 128 thicknesses were used. Composite sheets of gold and other metals were placed immediately over the radioactive source (ThC) and the ionization curves deter- mined in air with the same apparatus as before. Although a rough calculation seemed to show that the irregularities in the individual foils would be smoothed out enough to avoid masking any true increase in strageling, this result was not borne out by experiment. It was finally concluded that the increase in straggling observed was inainly, if not entirely, due to unevenness of the foils, and hence need not be gone into in detail here. In mica the increase in strageling was much the smallest, as was indeed to be expected. Fortunately, experiments on the straggling produced by Straggling of « Particles by Matter. 49 solid foils for low velocities of the « particles gave results which were not masked by irregularities of the foils. In these experiments the foils were placed 3 mm. from the middle of the ionization chamber (itself 1 mm. deep) in air at a pressure of roughly 17 cm. Reduced to air at normal pressure the distance from foil to centre of ionization chamber was therefore about ‘7 mm. Most of the foils used were made up of a few thicknesses of goldleaf. ‘lhe air equivalent of a single sheet was about ‘45 mm. when placed directly over the source ; ; when placed near the ionization chamber the air equivalent was about ‘28 mm. ‘The straggling of the « particles after passing through these foils was deter- mined in the same manner as before from the ionization curve in air. The results are given in Table II. The straggling is in millimetres, and the probable error is about 2 percent. ‘The third column shows the straggling observed when the foils are placed directly over the source, the steady increase with increasing number of leaves being mainly due to unevenness of foils. ‘The fourth column shows the increase in strageling at low velocity over that at high. This increased straggling is real and almost independent of the unevenness of the foils. Results with aluminium (1:0 mm. air equivalent) and mica (8°6 mm. air eq.) are also included in the Table, but with these foils the change observed is scarcely more . than the experimental error. Taste II. _ Straggling Foil | . . ‘ Blo. of Leaves , near lonization Straggling ae Difference. in Foil Gis hee near Source. ees de | 2:88 2:88 Fi | SEED ae nee | 3°30 2°92 38 et Cae emt | 3-65 | 3-01 64 SES ee | 4:00 3:28 72 Aluminium ...... | 3°08 | 3°01 ‘O7 1 SE eee eee 3°16 3°06 "10 These results show quite clearly the rapid increase of straggling near the end of the range. One gold leaf nearest the ionization chamber causes nearly twice as much strageling as three leaves immediately behind it. Hight more eaves Phil, Mag. 8. 6. Vol. 44. No. 259. July 1922. E - 50 Mr. G. H. Henderson on the placed behind these four again only slightly increases the strageling. The same result was also demonstrated in a slightly different manner. A foil made up of the four gold leaves already used was placed at different distances from the ionization chamber and the straggling determined from the ionization curves as before. ‘Phe results are shown in . Table III. The distances given in the first column are not ~ the actual distances from the foil to the centre of the ionization chamber, but are reduced to correspond with an ionization vessel containing air at atmospheric pressure. Taste ITI. Distance from Seah eat Tonization Chamber. ageing. “7 mm. 3°65 mm. ob ah 526, -12°9_,, 295, Here again it can be seen that the straggling increases rapidly near the end of the range. It may also be noted that the straggling (2°95 mm.) when the foil was 12°9 mm. from the ionization chamber is less than that (3:01 mm.) obtained when the foil was placed directly over the source. ‘lhe difference is less than the possible error, but such a change is to be expected. For the straggling due to unevenness of the foil should be less for low velocities of the « particles on account of the decrease in air equivalent of the foil. For the same reason the differ- ences given in the fourth column of Table II. are probably slightly too small. From the straggling observed with different types of a particles it was pointed out in § 2 that the strageling must occur within the last two or three centimetres of the range ; from the results with gold foils it seems that the straggling must be confined largely to the last few millimetres of the range. The increase of straggling due to gold foils placed near the end of the range, though clearly marked, is small. This is quite in accordance with the view expressed in § 3 that the strageling does not increase rapidly with increase of stopping power of the substance causing the stragoling, It must be remembered that the increase- observed is the Straggling of « Particles by Matter. ant increase over the strageling which would be produced by a layer of air equiv alent to the gold foil used. The ionization must be measured in a gas “such as air at a reasonable pressure, and hence we have the complication of the strageling due to the air between the solid and the ionization chaml Jer, and even in the chamber itself. ‘Chis sets a limitation to the amount of information to be obtained from ionization data. Accordingly the ionization experiments were not earried beyond the stage described, but it is hoped to push further the attack on the problem by more suitable methods. An isolated experiment with iron foils may be referred to before closing this section. A sheet of iron of about 4 cm. air equivalent was placed directly over the source and the ionization curve measured when the iron was magnetized parallel with and perpendicular to the direction of travel of the # particles. ‘The purpose of this experiment was to see if there was any change in the straggling due possibly to rearrangement or change of orientation of the electrons in the iron, a point which has been discussed by Flamm *. Alternate readings of the ionization with parallel and per- pendicular fields of about 100 gauss were made at various points along the range. No appreciable difference could be detected. It shouid be added that the iron was very uneven in thickness, and the consequent’ straggling was so large (about 20 mm.) that a small change in straggling might well have been masked. § 5. Summary and Conclusion. It has been shown in this paper that the straggling of a particles, as deduced from both ionization and counting experiments, is several times greater than that deduced from theory based on our present views of the mechanism involved in the passage of « particles through matter. It has been shown in a previous number of this Magazine that the effect of the calculated straggling can be adequately accounted for by the tailing off of the ionization curve at the extreme end of the range. The large additional straggling observed behaves quite differently. Hvidence has been given in this paper to show'that it increases very slowly with increase of molecular stopping power, and furthermore, that it all takes place within the last few millimetres of the range. Here we seem to be confronted with a behaviour of the a part cle which present theory is unable to explain. * Flamm, Wien. Ber. Ila. exxiv. p. 597 (1915). E 2 \ 2 ~ Dr. G. Green on Fluid Motion Evidence obtained from the Wilson photographs also- leads to the same idea, for it has been shown by Shimizu that the observed number of ray tracks which break up into two branches near the end of the range is much greater than the number deduced from probability considerations based on our present theory of atomic structure. It is noteworthy that this anomalous behaviour of the a particle occurs at low velocities, where practically no investigation of the scattering of « particles has been carried aut on account of the experimental difficulties of dealing with slow « particles. It is at higher velocities, where the theory of scattering put forward by Sir Ernest Rutherford has been so fully verified by experiment, that the most of the theoretical straggling takes place, and this straggling has apparently been accounted for. | In conclusion I wish to express my best thands to Professor Sir Ernest Rutherford for his kind interest and advice. I also wish to thank Mr. Crowe for the preparation of the radioactive sources. V. On Fluid Motion relative to a Rotating Earth. By Grorcr Green, D.Sc., Lecturer in Natural Philosophy in the University of Glasgow f. fi ese subject of this paper is at present one of consider- able interest to meteorologists. Papers by the late Dr. Aitken and also by the late Lord Rayleigh on the dynamics of cyclones and anticyclones have been followed by more recent papers by Dr. Jeffreys, Sir Napier Shaw, and others. Very few actual solutions of the equations defining atmospheric motions have been obtained. In the late Lord Rayleigh’s paper { attention is drawn to certain general hydrodynamical principles relating to the properties of rotating fluid which can be applied to “assist our judg- ment when an exact analysis seems impracticable.” The importance of the theorem regarding the circulation of the fluid in any closed circuit is clearly explained in its applica- tion to any actual fluid motion. In applying this theorem to fluid motion in the atmosphere, however, we must bear in mind that the motions with which we are concerned are not the actual motions of the particles in space but their motions relative to the Earth itself at each point of observation. . * Shimizu, Proc. Roy. Soe. xcix. p. 432 (1921), + Communicated by the Author. t Sc. Papers, vol. vi. p. 447, relative to a Rotating Karth. 53 One object of the present paper is to investigate the con- ditions under which the circulation theorem may be applied to atmospheric motions relative to the Harth’s surface ; or more generally to motions relative to any three rectangular axes which are themselves rotating about each other, with a fixed origin. In the later part of the paper one or two additional cases of motion of the atmosphere are discussed and the system of isobars corresponding to each motion determined. In view of the problems to be considered, we shall begin by specifying the system of rotating axes most convenient in dealing with fluid motion in the neighbourhood of any point of reference O on the EHarth’s surface. The axis OZ is drawn upwards along the apparent vertical at O, and line OZ continued downwards meets the axis of the Earth at a point QO’ which is taken as origin of coordinates. Then axes O’X and O’Y are drawn parallel to horizontal lines through the reference point O in directions due Kast and due North respectively. In the most general case to be considered the reference point O may be in motion relative to the Earth’s surface, and this involves also a motion of the origin O/ if point O moves either North or South. But the motion of O' corresponding to any moderate motion of O is very small, and for our present purpose we may regard the origin O' as a fixed point, very near to the centre of the Earth. We shall denote by («, y, z') the coordinates of any point referred to origin O', and by (a, y, z) the co- ordinates of the same point referred to parallel axes through O. This makes z'=z+R, where KR represents approximately the radius of the Earth. The components of the velocity of any particle relative to the axes at any instant are repre- sented by wu, v, w, and the angular velocities of the axes themselves, that is, of each two axes about the third, are represented by w,, w,, wz, respectively. We shall introduce the particular values of w,, wy, @, corresponding to a reference point O fixed in position on the Harth, or moving relative to the Earth, when we come to deal with special problems. Referred to the above system of axes, the equations of motion of any fluid particle take the form :— Du - OV 1 Op tego da SY Dv nA oes OP ; Pome a® Dw oy Lviap ay, nh =o ae ia: © ee @) 54. Dr. G. Green on Fluid Motion In these equations, 6), 0,, 8; represent the terms depending on the rotation of the axes themselves, being given by equations of the type 0,= —20,v+2o0,w—o.y+0,2' +0,0,) +0,0,2' —(0/ +o) a. (4) The function V(x, y, 2) represents the gravitational potential function. We have also a move) 0 Oli Pe CE ae 7. ae ee Ap Pe (9) The equation of igi of the fluid is then Ou Ov $) = == ().) ee pe te( Se + Se 4 Se (6) In applying these equations we treat the atmosphere as a perfect gas in which viscosity may be neglected. Circulation Theorem for Relative Motion. Consider now the theorem relating to the relative circula- tion. We have py (nde vdy+ waz) = By “dx + pi + yp e+), Dt (7) where g=w?+v?+w?, the square of the resultant relative velocity. By means of equations (1), (2), (3), the above ee may be rewritten in the form: py (udev dy +wd: ) =—(O,dx+, sdy+0,d2)—"P —aV + AG; Gaye (8) We can now integrate each term of this equation along any curve within the fluid from any point A to any point B. This integration gives the result, ‘B ay ie dp De (uda +vdy+wdz)=— nae dx + 0,dy +0@3;dz)— } — JA oe hei 7s —3q,° ; (9) and, if the integrations are applied to a closed curve relative to a Rotating Earth. 5D beginning and ending at the point A, we obtain = { (ude+vdy+wdz)=— \ (0,dx+O@,dy+O@3dz), (10) v5 «§ where the suffix 8 indicates that the integration is to be taken along a definite curve S. We have assumed in obtaining (10) from (9) that V is a single valued function of (wz, y, ¢), and that p isa function of p. It now appears that the rate of change of the relative circulation in any closed circuit which consists of the same fluid particles at all times is not zero unless, in addition to the above con- ditions, we have 00: 00;, 302301, 9% dz Boule ds. |) O66, LOM ha OF y. OF a a) When these conditions are not fulfilled, the relative vorticity does not move with the fluid itself, and if a velocity potential exists for a certain portion of fluid at a given instant, a velocity potential will not exist for that portion of fluid at a later instant. The first case of importance of the above conditions in relation to problems relating to the atmosphere is that in which the angular velocities w,, wy, @, of the axes are constants. Jn this case, the conditions given above take the form Pet ae meee Ou ov ow 12 fre Ores Oe ee Dz? seis ( ) where 6 represents an operator defined by Sao eee) 13) = Rig et ere . Pe ai ( y These equations have a solution of the form Uu (0) —— - =f (w,«+@,y+o,2), rae os (14) @, @y @, where f denotes any arbitrary function. If we draw an axis to coincide with the axis defined by the resultant of the three component rotations @,, ay, @:, then (@,v + wy + @22) is equal to ORcos¢@, where 2 is the resultant of (@;, @y, 2) and R is the line joining the origin to the point 2, y, 2. That is, wu, v, w are functions of p the perpendicular from the origin to a plane through the point (a, y, 2) perpen- dicular to the axis of the resultant rotation. 56 Dr. G. Green on Fluid Motion When the fluid is incompressible, and when a compressible fluid is moving in such a way that ou + SS + a IS Zero, a solution of a different type obtains. The solution in this case may be written in the form Uu=fi { (@yx—ary), (@24 — @,2) ei v=J2 {(wv—wzy), (w2%—@r2)},6. . « (15) PN RE Ayer in | where /;, /o, /s are arbitrary functions subject only to the condition oY + Oe ee =0(0. This solution includes as a ae’ 09° Oz | particular case any motion of rotation of the atmosphere as a solid about the axis of the Earth. The solutions which we have above obtained make it clear that the fluid motions relative to rotating axes in which the relative circulation moves with the fluid belong to a very restricted type. A relative motion, for instance, similar to that taking place in a free vortex, does not fulfil the con- ditions required for permanence of the velocity potential, and therefore no steady motion of this type could take place in the atmosphere—as has been assumed to be the case. The conditions which we have found to be necessary for the validity of the circulation theorem when the fluid motion is relative to rotating axes, may be obtained in a manner different from that employed above. Taking 3, H, Z to denote the components of angular velocity of a tuid ele- ment, and U, V, W to denote components of linear velocity of the element, each referred to fixed axes which coincide at instant ¢ with the instantaneous positions of the moving axes, we may derive the conditions from the equations employed by von Helmholtz in his papers on vortex motion :— Dei aU ol en So Dir = Oe oo ae 0s =(S- . OY 4s 02 73 a with two other similar equations. With &, , ¢ to represent the components of relative angular velocity of an element of fluid, referred to the moving axes, we have, B=E+o,; H=n+oe,; Z=l+o,; U=u-ayto,z; V=v—a,7+0,0; W=w-o0,«+ ory ; ~ and Din 2 = Oo 11@,6,. relative to a Rotating Earth. a7 By means of these relations we can readily transform (16) and obtain the corresponding equations for the rates of change of the circulation components of an element of fluid referred to the rotating axes ; in this way we find Dé Ou Ou Ou Ou Ov or) ‘am ae (5. + oy | 2 My. OU Ou One, Ov a Aw < tort bays) +8252 wo, (So 4 an (17) with the corresponding equations in 9 and ¢. Now the hydrodynamical theorem that relates to the permanence of a velocity potential for the motion of a given portion of fluid and the theorem of the permanence of the circulation of an element of fluid depend on equations (16). The equations which we have obtained for the relative circula- tions reduce to these equations exactly when the conditions expressed in (12) are fulfilled ; and these conditions must accordingly be fulfilled in order that the theorems referred to may apply to the relative motion, in the same way as they apply to the actual motion. Particular Cases of Motion Relative to the Earth, We shall now discuss one or two particular cases of fluid motion relative to the Earth, and we shall, to begin with, take the reference point O as a fixed point on the surface of the Earth at latitude ¢ degrees North. The angular velocity components @,, @,,@, have then the values 0, Ocosd, sind, respectively, where © represents the angular velocity of rotation of the Earth about its axis. If we now let a represent the perpendicular distance from any point z, y, < to the axis of the Karth, we can write the equations of motion (1), (2), (3), in the form Du ° yi eS OV! 1 0p >) ee Cet ee ae alae (138) BM sae bx Wane «yal: Op pp +22 sing -u ae ey a (19) Dw BeGuel: ap Seen — e t, == Sarees ———— e- = Z ) Dp 222 cos G - ¥ i 82? (20) where V'=V—1025%, V’ is, in fact, the potential function eee - 58 Dr. G. Green on Fluid Motion ! corresponding to apparent gravity, so that -< at point (v=0, y=0, z=0) is the value of —g at the reference point O. In applying the above equations to motion of the atmosphere, we may take . cea ee OV! | jn By) ee in the immediate neighbourhood of O in the region within which the value of apparent gravity may be regarded as constant in direction and amount. If we neglect a change of direction of one degree in g, our equations (18) to (21) would then represent conditions of motion within a radius of about seventy miles from point O. In order to render our equations suitable to represent circulations of air of diameter exceeding, say, 150 miles, we might employ the approximate values OMG Gt re 2 4ON a coterie 5 04,0 ROO have Be. oe ae (22) wherein we neglect the variation of g with height. If we exclude certain cases of motion relating specially to the tides, very few solutions of the above equations have been recorded. In order to make ourselves familiar with the types of fluid motion possible in the atmosphere it is of interest to examine all solutions which can be obtained having a bearing on meteorological problems. We, accord- ingly, take first the steady rotational motions of incom- pressible fluid under the force of gravity alone. We may take the boundary condition w=0 to apply at the surface of the Earth. A simple rotation of all the fluid about a vertical axis through O, with a uniform angular velocity w, would be represented by w=—=oy > t=+or; w=0. eee With these values in equations (18) to (2Q), it would be impossible to satisfy (19) and (20) simultaneously ; but the motion represented by Uu=—@(¥-—6z); v=o#r; “o—0,.: ee fulfils all the conditions contained in our equations, pro- vided 8 is given by a2 2Q, cos d 2 Bees D To! }-2 OFsima gh ote, Ce eee =) relative to a Rotating Earth. bY and the pressure p is given by P = —V' +4(w? + 20 sin p) {a2 + (y—B2)?} +0 (26) fe Nie The approximate value of V’ is + a(° a the pressure at the reference point O. In the motion indi- cated above each particle of fluid moves in a horizontal circle whose centre lies in the line (y=8z, x=0). This line lies in the meridian plane through point O and is inclined to the vertical at O at an angle @ towards the North, where tan@d=8. When a is very small in relation to © this in- clined axis is almost parallel to the axis of the Earth; and when o is large this axis comes almost to coincidence with the apparent vertical at O. With very large the motion described above corresponds very closely with a uniform rotation of the fluid about a vertical axis, as in the case of a simple forced vortex. Another case of motion of incompressible uid of interest in the same connexion is that represented by +2), and 9 1s oa i — oy toe; w= 20 cos. dh. 2, . (27) ae —V'+ 4 (o? + 200 sin ¢) (a? +y*) + 20? cos’ p . a? + a (28) as the equation showing the distribution of pressure. This motion differs from that first discussed in not being exactly horizontal. The plane of motion of each particle of fluid passes through the line OX, and is inclined to the horizontal plane XOY at an angle @ given by tan6=20 cos ¢/o. When the angular velocity of rotation w is very large com- pared with © cos@, the plane of motion of each particle is practically horizontal, and the motion then corresponds very closely with that of simple rotation of all the fluid as a solid about a vertical axis. When @ becomes small, on the other hand, the inclination of the plane of motion of each particle of fluid to the horizontal increases. The two motions, repre- sented by (24) and (27) respectively, are almost identical when @ is very large, and they differ entirely when @ is very small. It would be interesting to investigate the manner in which a fluid, such as water, subsides to rest from an initial condition of steady rotation about a vertical axis. The solution represented by (24), (26), would appear to be the exact solution for steady rotation of water in small scale experiments. 60 Dr. G. Green on Fluid Motion The motions considered above are motions of any incom- pressible fluid and do not indicate, except as approximations, conditions of motion possible in the atmosphere. One or two solutions of a similar type can be obtained which refer to incompressible fluid and accordingly represent motions possible in the atmosphere. Consider now the motion repre- sented by u=—o(y—62)3; v= 0 =O) ae The equations of motion of any element of fluid in this case are p28 242555, ee R Ox —20 sing .o(y—A2)=—% — $ log Pengo) SO eae eo e= g —h2-log p, Mice and these equations are satisfied provided the pressure system throughout the fluid is that indicated by ; d v+y? k log p=Qsin d. o(y—Bz)? —9 ap +2) +hlog po (33) where B=cot.¢, and po is the density of the fluid at the reference point O. The continuity equation is also satisfied provided we can neglect the term (ga/R)a(y— Bz). This condition limits considerably the extent of the region around O to which our solution is applicable, as stated earlier. Within the region to which the above applies the © isobars at the surface of the Earth run East and West, being determined by klog p=Qsing.ay?+klog po, . . . (84) which indicates a system symmetrical on the two sides of the Hast and West line drawn through reference point O. In this case the isobars become closer as we proceed North or South from point O. They are also parallel to the lines of flow of the fluid. The coefficient = has the value 1:5x 107° with the foot and the second as units; while 20sing has the value 1:03 x 1074 at latitude 45°. Certain cases of interest arise in which the terms containing may be neglected. For R relative to a Rotating Earth, 61 example, we may take the case of a uniform east or west wind over a considerable region. In this case eee Bae Me Oya tn tr. aN (3D) and the pressure distribution consistent with this motion is represented by klog p=2Qc (cos 6 z—sin g y) —gz +h log po, . (36) where py refers to the density of air at the reference point O. The isobars at the Earth’s surface are in this case a uniform system running due Hast and West. A similar case is that given by u=0, v=cer+d, w=0, .. . (87) which represents a wind towards the North, while the corre- sponding pressure distribution is that represented by klog p=Q sin ¢ (ca? + 2dx)—gz+klog py. . (38) The isobars are again a system of straight lines, but running north and south, and uniformly spaced when c=0. In a similar manner we find that U=c, V=c, we=0, . - . . (39) corresponds to a system of straight isobars represented by k log p=20 sin $ (qx—cyy) + 20 cosh .cye—gz + k log po. (40) The isobars are again lines of flow of the air, as in each case considered above. The case of motion corresponding most closely to a cyclonic or anticyclonic circulation is that discussed in an earlier paper *, represented by u=—o(y—Pz); v=er; w=0..., (41) In this case 20 cos wpeOsaran ce e Y AD) and the pressure distribution is that represented by klog p=3(@" + 2@Q sin $) {2 + (y—Bz)?} —gz +h log py. (43) J The term oe must be small in order that the continuity C= equation may be fulfilled. * Phil. Mag. vol. xli. April 1921 ; vol. xlii. July 1921. 62 Prof. G. N. Antonoff on the In each case considered we have only to replace 0 by an increased value ©' to obtain a motion for which the system of isobars travels eastward at a uniform speed (Q'’>92). Hach of the above solutions has been given to apply to an isothermal atmosphere, and in every case considered the fluid moves so that no element of fluid undergoes change of ; ee + ou = oy =( ateach point. Provided OL Oy, Oe this condition is fulfilled, any solution obtained for motion of an atmosphere, all at one temperature, can readily be trans- formed to suit an atmosphere in convective equilibrium (p=kp’), or one in which pressure is any given function of density. Thus, taking the motion represented by (41) above in an atmosphere in convective equilibrium, we have merely density. That is to replace log p and log py in (43) by re and wee respectively, all other conditions being unchanged. VI. The Breaking Stress of Crystals of Rock-Salt. By Prot. G. N. Antonorr, D.Sc.(Manch.) *. i a paper published in Phil. Mag. vol. xxxvi. Nov. 1918, I have developed a theory of surface tension under the assumption that the attraction of molecules is due to electrical or magnetic forces, or both. Instead of assuming a uniform field round the molecules as it is generally accepted acccording to Laplace, I accepted the view that the molecules act as electrical doublets, and from the theory of potential I deduced that the attraction between them must be inversely proportional to the 4th power of the distance, provided the distance between the doublets is large compared with their respective lengths. It was shown that the attraction between the doublets can be represented by an expression of the type ke dt ? where & is a constant, | the length of the doublet, and d the distance between them. In these calculations the magnetic forces were disregarded altogether, as the law of attraction between small magnets would be just the same, so that they could only have an effect on the value of &, * Communicated by Dr. J. W. Nicholson, F.RS. Breaking Stress of Crystals of Rock-Sult. 63 For the surface tension, the expression was given as k? 1 — | ¢ Cie tis or assuming qe =P where p is the number of molecules per ; ] unit volume, the expression for the surface tension « becomes N18 25/8 al ‘pp! 5 61 — 5s Ma the liquid, 6, that of the saturated vapour, and M the molecular weight. LOS NEE Thus wait (“5°*) re eRe iter oC) It was also shown that the internal pressure P can be calculated by the formula Bee ame ae oc ene) instead of p we may put where 6, is the density of = 1/3 ; or P=22 a5) ae aru dh wie) ae In other words, the intrinsic pressure can be calculated from the surface tension if the molecular weight of the liqnid is known. Thus for normal or non-associated liquids it should be possible to calculate the normal pressure from the value of the surface tension, It should be pointed out that the assumption made in the above theory, that the length of the doublet is small compared with the intramolecular distance, is not necessarily the right one. In the expression (2) this law is, however, eliminated, and the same expression is obtained for any law of molecular attraction. ‘The figures for P obtained from the above expression agree with those from indirect evidence. How- ever, experimentally it is not possible to determine P directly, owing to the mobility of the particles of liquids which always adjust themselves so that the molecular pressure is inappreciable. It is not so in the solid or crystalline state, in which the particles have a definite orientation, and where the internal pressure can be determined by a direct experiment. It is sufficient to apply to a crystalline body such a weight as would overcome the attraction of the molecular forces and cause the disruption of the body. The force applied is not 64 Prot. G. N. Antonoff€ on the necessarily the same in all directions, and it is therefore necessary to specify the direction in which it is to be applied. The question arises now whether it is possible to calculate the surface tension of a solid body. For the solid state there is no direct method of determining the surface tension, all methods used for liquids being inapplicable in this case. Some attempts were made to estimate the surface tension of solids from indirect evidence. © For example, Ostwald * and Hulett t calculated the surface tension of some calcium and barium salts on a basis of a certain theory from the solubility data. The figure given for the latter is about 4000 dynes percm. From the point of view of our theory, it seems possible to calculate the surface tension by the use of formula (2) by determining experi- mentally the internal pressure per square cm. of the cross- section, if the molecular structure of the substance in the crystalline state is known. At the present time the X-ray analysis throws a light on the above question. | - For example, according to W. H. and W. L. Bragg, the erystal of rock-salt consists of charged ions situated at regular distances from one another. Such a case is somewhat different from the one discussed in my paper (loc. cit.). Here it is necessary to assume that l=d, where / is the length of the doublet and d the distance between them, under which conditions the ordinary inverse square law must hold true. The attraction between the charges in a row is equal to e*k dq? where ¢ is the elementary charge, and the value of & is the sum of a series L—$+4—}43—...=0°6931. Assuming that the adjacent rows have no effect upon the charges, the expression for the surface tension is of the form 2 [a=ekp, where p=number of particles per unit volume. For the normal pressure the expression will be : P=khe*p*, * Zeit. Phys. Chem. xxxiv. p. 503 (1900). + Zeit. Phys. Chem, xxxvii. p. 386 (1901). Breaking Stress of Crystals of Rock-Salt. 65 For rock-salt the density = 2°15, M164 ¥5S'o: x. 10-73, — 2 2 yA x TO", Assuming that & is approximately=0°7, the tension is BS cOrt aed 10% 29 x 10-28 x, OT x 2°24. « 107? =3500 dynes per cm. (approximately). The figure obtained is of the same order of magnitude as figures derived by Ostwald and Hulett (loc. cit.) for barium salts. The normal pressure P would be accordingly P=98'7 x 10° dynes per square cm. It is interesting to see now how far the above results agree with the experimental evidence. An experiment was performed as follows :— I took a good specimen of rock-salt crystal and I cut a prism of the section about 15 square mm. and about 2-3 cm, long. I used a suitable cement to hold the piece from both ends, and by applying a suitable weight produced a rupture of the crystal into two halves. Measuring the cross- section of the rupture accurately, I calculated the weight required to produce the rupture per square cm. I have repeated the experiment many times with different samples of rock-salt. I£ the crystal is well formed, the agreement between individual experiments is fairly good. In one series of experiments, I cut the prisms so as to have several samples cut parallel to the three principal axes. I have done the experiment with three pieces for each direction. For one direction I obtained : 89 lb. per cm, BOS 45 1155 83 39 33 In the other two directions the results were identical. In some cases it happened that the rupture took place under a much smaller weight. This, however, could be attributed either to some faults in the structure of the crystal, or to some other disturbances. Such measurements were simply disregarded. Taking as the average value 91°7 lb., or 41°5 kegrs. per square cm., one can calculate the inward pull per row of unit length. Phil, Mag. 8. 6. Vol. 44. No. 259. July 1922. F 66 Breaking Stress of Crystals of Rock-Salt. This is obtained by dividing P by p'*, and becomes P _ 415000 x 981 aa ayn Bap Bert Fert ee If the attraction between the charges in a row is not appreciably influenced by the adjacent rows, this value will represent the accurate value of surface tension in the direction coinciding with the vertical axis. If the field is symmetrical in all three directions parallel to the main axis, this figure will characterize the surface tension of rock-salt in all three directions. The symmetrical structure indicated by W. H. and W. L. Bragg is in accord with the experiment in this sense, but on the other hand, the above figure is about 1000 times less than one would expect. One could expect such a small value if the salt consisted of molecules with a very small polarity situated at large distances from one another. But in such a case the force in the direction coinciding with that of the doublet would have to be twice that in the perpendicular direction. ‘This, however, is not the ease. In my paper (loc. cit.) it was shown that in a case of small doublets the adjacent rows have practically no influence on the attraction between the doublets. However, in a case of charges situated at regular distances from one another, such seems not to be the case. If the charges influence one another, one can expect the forces to be weaker in the middle of the substance, and much bigger at the surface where the above effect is only one- sided. It is therefore probable that the above figure 1:4 dynes per cm., although quite characteristic for the substance, is not the actual value of the force in the surface layer. The calculation of these effects is not easy, owing to the fact that one has to deal with a very slowly converging series. I[ satisfied myself that these influences may be appreciable, but I do not see clearly at the present time whether they can account for the weakening of the forces about 1000 times, or even more. 6 Featherstone Buildings, High Holborn, London, W.C. 1. VIL. The Measurement of Chance. By Norman Campsett, Sc. D.* Summar Ys T is maintained that the chance of an event happening is always a physical property of a system, measured by a process of derived measurement involving the two fundamental magnitudes—number of events and number of trials. Chances are not measurable by a process of fundamental measurement. But the calculation of chances is analogous to fundamental measurement. It is usually theoretical, and is valuable only in so far as the calculated chances are confirmed by measurement. When a proposition concerns a system characterized by a chance, it may sometimes (but by no means always) be regarded as having a definite probability determined by that chance. The probability of propositions which do not concern systems characterized by chances has nothing to do with chance. Ll. It is generally recognized that there are two kinds of “probability.” There is (1) the probability (of the happening) of events, and (2) the probability (of the truth) of propositions. tymologically the term belongs more properly to the kind second of probability, and it will be confined to that kind in this paper. For the first kind, the term “chance,” often used in some connexions as a synonym of probability, is available. Accordingly we shall speak throughout of the chance of an event happening and of the probability of a proposition being true. Various opinions have been entertained concerning the relation between chance and probability and between the methods of measuring them. Some have held that chance, some that probability, is the more fundamental conception, and that the measurement of the less fundamental depends on that of the more fundamental conception. Others have held that only one of the two, or neither, is measurable. The conclusion towards which this paper is directed is that chance, in the sense primarily important to physics, is a phy- sical property measurable by ordinary physical measurement. This view is similar to that held by Venn; indeed, it is * Communicated by the Author. 2 68 Dr. Norman Campbell on the probably the view that Venn would have held if he had ever considered the nature of physical measurement. But the further view often attributed to Venn, though it is doubtful whether he actually held it, that probability is always measurable in terms of chance—this view will not be upheld, but, so far as it is discussed at all, will be combated. Many of the ideas and terms used in the discussion are explained more fuliy in my ‘ Physics,’ to which references are made by the letter P. In fact, this paper may be regarded as a substitute for pp. 168-183 of that book, some of the difficulties of which are avoided by the alternative method of treatment. However, I should like to add that I do not accept any of the criticisms that have been directed against those pages by others. | 2. Suppose we are presented with a pair of dice, and asked what is the chance that when one of them is thrown it will turn up six. The answer may be different for the two dice. If one of them is accurately cubical with its centre of mass accuraiely at the centre of the cube, while the other has corners and edges variously rounded and is loaded so that the centre of mass is appreciably nearer one face than another, then the answer will be different. On the other hand, if in all respects the dice are the same, then the answer will be the same; even if they are both inaccurate in form and both loaded—the inaccuracy of form and the loading being the same—the chance that they will turn up six will be the same. This chance is something uniformly associated with and changing with the structure of the die, just as is (say) the electrical resistance. This uniform association of the resistance with the other characteristics of the die is what we assert when we say that the resis- tance is a physical property of the die, and accordingly the chance of turning up six is a physical property as much as tle resistance. 7 | Moreover, the chance is measured by essentially the same process as that by which the resistance is measured. Resist- ance is measured (in its original meaning) as a derived magnitude by means of a numerical law (P. Ch. xiii.), We place two electrodes in contact with opposite faces of the die, and measure the current which flows through it when measured potential differences are maintained between the electrodes. We then plot current against P.D., and find we ean draw a straight line through (or more accurately among) the resulting points. The fact that the graph isa Measurement of Chance. 69 straight line passing through the origin shows that a numerical law of a certain form holds, and therefore that the die is characterized by a single definite magnitude, which is what we mean by resistance ; the slope of the line tells us the numerical value of this magnitude. When we proceed to measure the chance of turning up six, we make several groups of trials, measure in each group the number of trials and the number of those in which six turns up. We plot these two fundamental magnitudes against each other, and find that a straight line can be drawn through (or among) the points. The fact that the graph is straight and passes through the origin tells us that the die is characterized by a definite magnitude, which is what we mean by chance ; the slope of the line tells us the numerical value of this magnitude. 3. The resemblance is exact in all essentials. But as the conclusion that chance is an ordinary physical magnitude does not seem to be universally accepted, some objections may be considered. The first may be (though I am not sure that it will be) raised by those who denounce the “frequency theory” of probability. They might say that, though the derived magnitude, estimated in the manner described, isa true or approximate measure of the chance, yet it is not what is meant by the chance—that is something much more abstruse. Such an objection can only be met by stating more clearly what is asserted, and recognizing any difference of opinion that remains as insoluble. What I assert is (1) that all chances determined by experiment are determined by a relation between frequencies, and (2) that chances are important for physics only in so far as they represent relations between frequencies. Few examples can be cited in support of (1), for chance in physics is usually a theo- retical and not an experimental conception ; but it may be suggested that anyone who proposed to attribute to the chance of a given deflexion of an a-ray in passing through a given film any value other than that determined by fre- quency, could convince us of nothing but his ignorance of physics. In support of (2) it may be pointed out that the chance, which is such an important conception in the statistical theories of physics, enters into the laws predicted by those theories only because it represents a relative frequency. 70 Dr. Norman Campbell on the 4, A second objection may be based on the fact that the straight line has to be drawn among and not through the experimental points. It may be readily admitted that this fact shows that the chance cannot be estimated with perfect accuracy. but there is also some uncertainty in determining the resistance ; and since I am concerned only to enforce the analogy between chance and resistance, the admission is innocuous. If it is urged that this uncertainty shows that the derived magnitude cannot be the chance, because chance is something to which a numeral may be attached with mathematical accuracy, then it is replied (as in answering the first objection) that such a chance, to which no experi- mental error is attached, is something totally irrelevant to physies. . | But the objection may be put in a less crude form. It may be urged that, in the matter of experimental error, there is a fundamental difference between resistance and chance. For in the latter, but not in the former, the error is something essential to the magnitude ; we can conceive of a resistance measured without error, but not of a chance measured with- out error ; if all the points lay accurately on the line, then the magnitude measured by its slope would not be a chance. Again, there is a simple relation between the average error about a point on the ‘‘chance”’ line and the co-ordinates of that point ; while in the “resistance” line the relation is much more complex, and depends on the exact method of measuring the current and potential. All this is quite true, and would be important if we were considering the theory of chance or of resistance. There is a great difference in those theories; we suppose that the ‘‘real” Ohm’s law holds between the real and not the measured magnitudes of the current and potential, while there is no real magnitude involved in the chance relation. But we are not considering theory but experiment; I am only asserting that chance is an exper!l- mentally measured magnitude. The fact that the errors in the two cases are differently explained does not affect the fact that there are errors in both cases, and that the problem of determining the derived magnitude in spite of these errors is precisely the same. 5. Asa third objection it might be urged that the two measurements are not really similar, because the chance is not really determined by the slope of the line, but by the ratio of the two numbers when they are sufficiently great. Here is a misconception which it is important to correct. If we know that the happening of the events is determined Measurement of Chance. th by chance, then it is true that we need only plot one point on the line ; and the distribution of the ‘errors is such that the relative error of a determination from a single point is less the greater the number of trials involved. We shall group all our observations together, so as to imake their totai number as greatas possible. But similarly, if we know that the material of the die obeys Ohm’s law, one observa- tion is sufficient to determine its resistance ; and the accuracy of the determination will be greatest if we choose the measuring current within a certain range. An even closer parallel would be obtained if we took in place of resistance the derived magnitude, uniform velocity. If we knew that the velocity was uniform, we should choose our time and distance as great as possible, and determine the velocity from this single pair of values without troubling to plot smaller values. But in order that determination by a single point should be legitimate, we must know that the events really are determined by chance, and the only test of chance is that, when a series of points are plotted in the manner described, the only regularity discoverable in them is that they lie about the straight line. Their distribution about that line must be random. Thus, to take Poincaré’s excellent example, if the trials were made by selecting the first figure of the numerals in a table of logarithms in the conventional order, and the events were the occurrence of the figure 1, the plotted points would lie on the whole about a line with a slope of 1/10. But a regularity of distribution about that line would be apparent; we should have a series of points all lying above the line followed by a set all lying below. If, on the other hand, we took the last figure of the numerals, no such regularity would be apparent; the distribution of the points about the line would be random ; the events would be dictated by chance. It is of the first importance to insist that in measuring a chance we are picking out the only regularity that we can find in some sequence of phenomena, leaving a residuum which is purely random. Randomness is a primary con- ception, incapable of further definition; it cannot be explained to anyone who does not possess it. It is based, I believe, on observation of the actions of beings acting consciously under free volition ; and it is subjective in the sense that what is random to one person may not be random to another with fuller knowledge (P. p. 203). There are certain forms of distributions that are random to everybody ; it is this common randomness, objective in the sense in which all the 72 Dr. Norman Campbell on the subject-matter of science is objective, that is the characteristic of the objective chance which is physically measurable. Chance is applicable only to events which contain an element which is wholly and completely random to everybody *. 6. We shall then base our further discussion on the assumption that any physically significant chance (of the happening of an event) is a measurable derived magnitude, a property of the system concerned in that event, determined by a linear numerical law relating the fundamental magni- tudes, number of events and number of trials. It is thereby implied that the “errors” from the law are random, for otherwise the law would not be linear. The definition of chance as the limiting ratio of the fundamental magnitudes as they tend to infinity is identical with that given, if it is known as an experimental fact that the magnitudes of the errors fulfil certain conditions which need not be discussed in detail here; these conditions are not inconsistent with the randomness of the errors. Chanee as a fundamental magnitude. 7. Another important question may be raised, again suggested by the analogy with resistance, Resistance means the derived magnitude defined by Ohm’s law. But actually resistance is measured nowadays, not as a derived, but asa fundamental magnitude, in virtue of the Kirchhoff laws for the combination of resistances in series and parallel f. Can chance, though meaning the derived magnitude, be measured independently as fundamental ? In order that a property may be measured as a funda- mental magnitude, it is necessary that satisfactory definitions of equality and of addition should be found (P. Ch. x.). In addition, some numerical value must be assigned arbi- trarily to some one property, which with all others can * In this sense the last figure of the logarithm is not wholly dictated by chance; for we know that there must be some regularity in the distribution of the points about the straight line, even if we cannot say exactly what it is. In the strictest sense, therefore, there is no such thing as the chance of the last figure being 1. But there are events which are, at present at least, wholly dictated by chance in this sense, e.g. the distintegrations of a radioactive atom. Here I do not think anyone has imagined what kind of regularity there can be, except the falling of the plotted points about the straight line which determines the chance. t+ Ultimately measured, that is to say, hy the makers who calibrate our resistance boxes. In the laboratory we use a method which is essentially that of judgment of equality with a graduated instrument. Measurement of Chance. 73 be compared by means of these definitions. Since we take the meaning of chance to be that of the derived magnitude, the definitions will be satisfactory if they are in accord- ance with the derived process of measurement; but we shall not succeed in establishing an independent system of fundamental measurement, unless the definitions are such that they can be applied without resort to that process. The arbitrary assignment is usually made by attributing the value 1 to the chance of an event which always happens as the result of a trial. The only question that can arise here, namely whether all other chances can be connected with this chance by addition and equality, will be considered presently. The definition of addition presents no difficulty. The chance of A happening is the sum of the chances of x, y, 2,... happening if, wv, y, z, ... being mutually exclusive alternatives, A is the event which consists in the happening of either z or y, or z,..... This proposition is introduced in all discussions of chance, but it is often introduced as a deduction and not as a definition. The inconsistencies which result from such a procedure are discussed in P. pp. 174, 184, 185. As a definition, it is satisfactory in our sense, for measurements by the derived method would show that, in such conditions, the chance of A is the sum of the chances of , y, 2,...,and yet it does not presuppose such derived measurements. If the points in the derived measurement lay on the straight line, this result would be a direct consequence of the definition of the derived magnitude ; but since they do not, it can be deduced from that definition only if some assumption about the distribution of the errors is made. The assumption that the errors are random would probably suffice if randomness could be strictly defined; since it cannot, the agreement of the proposed definition of addition with the results of the derived process of measurement must be regarded as an experimental fact. The definition is thus precisely analogous to that used in the fundamental measurement of resistance, namely that resistances are added when the bodies are placed in series. The definition of equality is much more difficult ; in fact, it is the stumbling block of many expositions of the measure- ment of chance. [For resistances we can say that bodies are equal if, when one is substituted for another in any circuit, the current and potentials in that circuit are unchanged ; that definition does not involve a knowledge of Ohm’s "law and of the derived measurement. The only attempt at an analogous definition for chance, of which I am aware, is that based on the principle of sufficient reason; chances 74 Dr. Norman Campbell on the are said to be equal when there is no reason to believe that one rather than the other will happen as the result of any trial. But what reason could there be for such a belief based on experiment? No a priori principle can determine the property of a system, which is an experi- mental fact ; we cannot tell whether a die is fair or loaded without examining it through our senses. The only experi- mental reason I can conceive for believing that one event is more likely to happen than another is that it has happened more frequently in the past. But if an attempt is made to define ‘“‘ more frequently ” precisely, the judg- ment of equality is inevitably made to depend on the derived measurement, and the fundamental process ceases to be independent of it. This dependence is often con- cealed by the use of question-begging words. Thus, the principle of sufficient reason may be reasonably held to decide that, in a perfectly shuffled pack of cards, the chance that the card next after a heart is another heart is equal to the chance that it isa club. But if inquiry is made what is meant by a perfectly shuffled pack and how we are to know whether a pack is or is not perfectly shuffled, I can seen no answer except that it is one in which a club occurs after a heart as often as another heart. But, of course, to define perfect shuffling in that way is to admit that the criterion of equality is based upon the derived measurement of ‘frequency’ *. I can find no proposed definition of the equality of chances that is both applicable to experimental facts and independent of frequency ; and I conclude, therefore, that there is not for chance, as there is for resistance, a fundamental process of measurement independent of the derived. 8. But there is a further difference to be considered. Even if equality of chance could be defined independently, there would still be many chances (and those some of the most important) which could not be connected with the unit by the relations of equality and addition. Any resistance is equal to the sum of some set of resistances such that the sum of another set of them is equal to the unit or to the sum of some set of units. The analogous proposition about chances * It is not always realized by those who calculate card chances in great detail that in actual play, even among experienced players, the shuffling is so imperfect as to distort very seriously the chances of such events as the holding of a very long suit. Measurement of Chance. 75 is not true, if the chances are always experimentally deter- mined. Consider, for example, the disintegration of a radioactive atom within a stated period. There are only two alternatives: the atom does distintegrate, or it does not. The sum of the chances of these two events is 1, but the chances of the two are not in general equal. And neither of them can be shown experimentally to be the sum of the equal chances of other events such that the sum of some other set of those chances is equal to the unit. The definition of unit chance together with definitions of equality and addition would never permit us to determine such chances ; they can only be determined by derived measurement. 9. Chance is therefore not capable of fundamental measurement. Nevertheless the principles of fundamental measurement are important in connexion with chance, because they are involved in the calculation of chances. When we calculate a chance we always assume that it is measurable by the fundamental process. Thusif we calculate the chance of drawing a heart from a pack, we ¢ argue thus :— The chance of drawing any one of the 52 cards is equal to that of drawing any other. The chance of drawing one of the 52 cards is, by the definition of addition, the sum of the chanees of drawing the individual cards, and, by the definition of unit chance, itis 1. Consequently the chance of drawing any one card is 1/52. Butthe chance of drawing a heart is the sum of the chances of drawing 13 individual cards ; it is therefore the sum.of 13 chances each equal to 13/52, i 2.e.1/4. The calculation is perfectly legitimate, so long as we know (1) of how many individual events tlie event under consideration (and any other event introduced into the argument) is the sum, and (2) that the chances of these individual events are equal. (1) does not depend on the derived system of measurement, but it does involve a very complete knowledge of the event under consideration; (2), if it is an experimental proposition at all, must depend upon derived measurement. The calculation is often made when (2) is not experimental, and when there is no direct know- ledge of (1); it is then purely theoretical, and the only legitimate use that can be made of it is to confirm or reject the theory by means of a comparison of the calculated chance with that determined experimentally by the derived measure- ment. The fact remains that true chance, the property of the system, is always and inevitably measur ed by the derived process and not by the fundamental. 76 Dr. Norman Campbell on the Chance and Probability. 10. It remains to consider very briefly what connexion, if any, there is between the chance of events and the proba- bilities of propositions. Probability is usually admitted to be an indefinable conception, applicable to propositions concerning which there is no complete certainty, and roughly describable as the degree of their certainty. It appears to me one of those conceptions which are the more elusive the more they are studied ; | am quite certain that I do not understand what some other writers mean by the term, and am not at all certain that I can attach a perfectly definite meaning to it myself. The observations that I can offer are therefore necessarily tentative. But it is clear, at any rate, that probability is not a property of a system and is not physically measurable: any propositions connecting it with chance must depend ultimately on fundamental judgments which can be offered for acceptance, but cannot be the subject of scientific proof. There are two kinds of propositions the probability of which may plausibly be connected with chance ; and they naturally can apply only to systems that are characterized by chances. Of the first kind the following is typical :—This die will turn up six the next time it is thrown (or on some other single and definite occasion). Here (cf. P. pp. 192-200) it seems that, if the proposition is really applied to a single occasion only, the probability of the proposition must be that characteristic of absolute ignorance ; for the assumption that anything whatever is known of the result of a single trial is inconsistent with the experimental fact that the result of any one trialis random. The only exception occurs when the event is one of which the chance is so small (or so great) that the happening of it (or failure of it) would force us to revise our estimate of the chance or to deny that there was a chance at all. Of such coincidences, in systems of which the chance has been well ascertained, the assertion that they will not (or will) occur may be made with the certainty that is characteristic of any scientific statement. There is no probability. On the other hand, it is very difficult to be sure that only a single trial is contemplated. For when such statements are important, there is alwaysa clear possibility of a consider- able number of repetitions of the trial. If this number is so great as to permit a dermination, by derived measurement, of the chance of the event within some limits relevant to the Measurement of Chance. “a problem, then it will be found by examination of the use of such propositions that their importance depends simply on the value of that chance. If that chance is greater than a certain value, the proposition will be true for the purposes concerned ; if it is less, it will be false. I cannot myself ever find in such propositions any meaning which is not contained in the proposition :—The chance that the die will turn up six is greater or less than some other chance. Accordingly again, there seems no room for a probability which is distinct from chance. 11. Of the second kind of proposition an answer to the following question may be taken as typical :—I have two dice, of which the chances of turning up six are unequal. I throw one, but I do not know which. It turns up six. Which of the two dice have I thrown? Here again (P. pp. 185-192), if the question is asked of a single throw, it seems to me that the only possible answer is simply, I do not know; except, as before, if the throw would be a “ coincidence ’’ with one die and not with the other. For, once more, if the events concerned are really characterized by chances, it is inconsistent with the statement that they are so characterized to assert that, at a single trial, the result, if compatiole with either of the two ““causes,’ may not happen as the result of either of them. If, on the other hand, the throws are repeated (while it is certain that the same die is always used), and if they deter- mine the chance of one die rather than that of the other, it is clearly certain that this die, and not the other, is being used ; «a die can be identified by its chance as certainly as by its resistance or any other physical property. But intermediate between these extremes, there certainly seem to be cases in which, though the evidence is not sufficient to enable us to assert definitely which die is being thrown, we begin to suspect that it is one and not the other. ‘Lhe possibility of such a state of mind arises from the fact that there is necessarily a finite period during which the evidence is accumulating; it does not arise when, as in the usual determination of resistance, the evidence is obtained all at the same time. And our suspicion will increase generally with the “ probability ”’ as estimated by the well- known Bayes’s formula for the probability of causes, In this case it appears to me that there is such a thing as probability, determined by but distinguishable from chance, and applying to a proposition, and not to au event. But I can find no reason to believe that this probability is 18 The Measurement of Chanee. numerically measurable in accordance with Bayes’ or any other formula. But in most cases where an attempt is made to apply a probability of causes, the condition is not fulfilled that it is known that the same die is always used. If that condition is not fulfilled, the probability, according to orthodox theory, depends on certain a prior probabilities which are not chances. The problem then ceases to be one of the connexion between chance and probability, and thus falls without the strict limits of our discussion. 12. But itis necessary to transgress those limits for one purpose. It has been often urged by philosophers that probability is characteristically applicable to scientific propositions, which are to be regarded, not as certain, but only as more or less probable. If this be so, the con- ception of chance, being a scientific conception deriving its meaning from scientific propositions, must be subsequent to the conception of probability, and the order of our discussion should have been reversed. Of course I do not accept the philosoph:eal view, and perhaps it will be well to explain very briefly why I reject it. Doubtless there is a sense in which scientific propositions are not certain ; but in that sense no proposition is certain, so long as its contrary is comprehensible. For if I can understand what is meant by a proposition, | can conceive myself believing it. Iam not perfectly certain ae that Ohm’s law is true, or that (o+a)?=a?+2ax+a?: I can conceive myself disbelieving either. If I were Focestl to say which I believe mere certainly, I should choose Ohmn’s law ; for I could give a much better account of the evidence on which I believe it. A mathematician, of course, would mike the opposite choice. But it appears to me useless to com- pare the “certainties”’ of two propositions when they are of so different a nature that the source of the uncertainty is perfectly different. If a proposition 1s as certain as any proposition of that nature can be, and if nothing whatever could make it more certain, then it seems to me misieading | to distinguish its pr obability from certainty. Now, fully established scientitie propositions are ecrtain in this sense. They are uncertain only in so far as they predict If in asserting Olim’s law, I mean (and I think this is my chief meaning) that it appears to me a perfectly complete and satisfying interpretation of all past experience and that other prone appear to share my opinion, then Ohm’s law is Ignition of Gases by Sudden Compression. 79 perfectly certain, or at least as certain as any mathematical or logical proposition. Onthe other hand, if it is meant that the law will be to me and to others an equally satisfying interpretation of all future experience, then [I am _ not absolutely certain ; I am only as certain as I can be about anything in the future. And it must be noticed that nothing can make me more certain. If I were predicting something about a single future occasion, I might in the course of time become more certain; for that future occasion might some day become past. But if, as in the case of a scientific law, I am predicting something about all future experience, then, since the future is indefinite, no amount of additional experience, converting finite portions of the future into the past, can make me more certain; for there will always remain as much future as before. Such uncertainty as there is in the proposition is inherent in its nature; if it were absolutely certain, it would not be the same proposition, VIII. EHvperiments on the Ignition of Gases by Sudden Compression. By H. T. Tizarp and D. Rh. Pye*. [Plate I.] i. a a previous paper J, it was shown that when a mixture of a combustible gas or vapour with air was suddenly compressed, explosion might take place after an interval the duration of which depended on the temperature reached by the compression. It is known that below a certain tempera- ture, called the ignition temperature, no explosion, and no very appreciable reaction, takes place under these conditions ; and the experiments referred to showed that just above the ignition temperature, the delay before explosion occurs may be of the order of one second in certain cases, while—in the case of hydrocarbons and air—the delay at a temperature some 50° above the ignition temperature was very small. It was pointed out that the observed ignition temperature must not only depend on the properties of the combustible substances, but also on the conditions of experiment, and particularly on the rate of loss of heat from the gas at the * Communicated by the Authors. + H. T. Tizard, “The Causes of Detonation in Internal Combustion zines.” Proceedings of the N.E. Coast Institution of Engineers and Shipbuilders, May 1921, 80 Messrs. H. T. Tizard and D. R. Pye on the ignition temperature. The fact that this has not been fully taken into account previously seems to account in some measure for the differences in the results obtained by other workers. It was further shown that the period of slow combustion before explosion took place also depends on the properties of the combustible substance, and a theory was briefly developed connecting the “delays” observed at different temperatures with the effect of a risein temperature on the rate of combustion, 7. e. with the so-called tempera- ture coefficient of the reaction. The object of the experi- ments described in this paper was to test these theories quantitatively, and to attempt to deduce from the results the temperature coefficients in certain typical cases. The mea- surement of the temperature coefficients of simple gaseous reactions is of considerable importance in connexion with the theory of chemical reactions, for one of the great difficulties in the development of theory hitherto has been the fact that most gaseous reactions have to be investigated under conditions which are complicated by the disturbing influence of solid catalysts or of the walls of the containing vessel. Gaseous reactions which occur on sudden com- pression are free from this complication, for the walls of the containing vessel are much lower in temperature than the gas; by quantitative measurements of the rate of loss of heat near the ignition temperature, and of the delay before explosion occurs, it therefore seems possible to gain some real insight into the mechanism of homogeneous gas reactions. Experiments of this nature also havea considerable practical interest for the development of internal combustion engines, for, according to our views, the tendency of a fuel to detonate at high temperatures depends not only on its ignition temperature but also on the temperature coefficient of its reaction with oxygen. Previous experiments on the ignition of gases by sudden compression have been made by Falk, at Nernst’s suggestion (J. Amer. Chem. Soe. xviii. p. 1517 (1906), xxix. p. 1536 (1907) ), and by Dixon and his co-workers (see Dixon, Brad- shaw & Campbell, Journ. Chem. Soc. 1914, p. 2027; and Dixon & Crofts, p. 2036). Nernst first put forward the view that at the ignition temperature the evolution of heat due to the reaction was just greater than that lost to the sur- roundings; but this suggestion has not hitherto been carried further, since no previous workers have attempted. to measure the rate of loss of heat near the ignition tempera- ture. Further, previous work has been mainly confined to the measurement of ignition temperatures of mixtures of Ignition of Gases by Sudden Compression. 81 hydrogen, oxygen, and an indifferent gas. In such experi- ments the interval which occurs at the lowest ignition temperature between the end of the compression and the occurrence of ignition is very small; under these conditions an apparatus of the kind used by Nernst and Dixon gives fairly satisfactory results. It is not well suited, however, for experiments with other gases, such as the hydrocarbons, when there may be an appreciable delay before ignition occurs. In such cases it is of great importance to ensure that the cylinder in which the compression is effected is as gas-tight as possible, and that the piston is held rigidly in position at the end of the compression stroke. II. The apparatus used for our experiments was originally designed and built by Messrs. Ricardo & Co. with a view to determining the temperatures of spontaneous ignition of various fuels used in internal combustion engines under conditions which correspond closely with those obtaining in an engine cylinder. Fig. 1 shows diagrammaticaily the arrangement of the mechanism. A very heavy flywheel A rotates quite freely on the shaft B, and is kept spinning by an electric motor at about 360 R.P.M. The shaft B carries between bearings the crank D, and outside one bearing, the internal expanding clutch ©, which can engage with the flywheel rim. The piston E moves vertically in the jacketed cylinder F, which has an internal diameter of 44 inches and can be raised or lowered bodily in the heavy cast-iron casing of the apparatus when the compression ratio is to be altered. The length of stroke of the piston is 8 inches, and its motion 1s controlled by the two hinged rods G and H of which the latter is carried on a fixed bearing at K. The hinge L is Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. G - 82 Messrs. H. T. Tizard and D. R. Pye on the linked up with the crank pin by the compound connecting rod N. That part of the connecting rod attached to the crank pin is tubular and contains the sliding rod M attached to the hinge L. A clip O carried on the sleeve can engage with a collar on the inner rod and hold the latter rigid in the tube. With the connecting rod I-cked as one link, the crank is rotated by hand for setting the piston in its lowest position. When a compression has to be made, tle clutch is suddenly expanded by a hand lever while the flywheel is running at high speed, clutch and crank are carried round with the flywheel, and the toggle joint ELK is straightened until the hinge L lies on the vertical line between the piston centre and hinge K. At the moment when L is vertically over K it comes up against a leather pad, and a clip comes into action which holds it in this position. At the same moment, too, the clip O releases the two parts of the com- pound connecting rod, so that while the two rods G and H are held in the vertical position to take the large downward thrust of the piston when explosion of the compressed mixture occurs, the crank, clutch, and flywheel are free to go on rotating, and the shock due to destruction of the momentum of the moving parts is reduced to a minimum. The initial temperature of the gases in the cylinder can be varied by means of a water jacket round the cylinder, and the variation of pressure during and after compression is recorded by means of an optical pressure indicator of the Hopkinson type. For the purposes of the present experiments it was necessary :— (1) To make the compression space above the piston absolutely air-tight so as to eliminate all pressure drop due to leakage. (2) To arrange an accurate timing gear upon the indicator so that pressure-time records correct to about one per cent. could be obtained, (3) To insert some kind of fan in the compression space so that the effect of varying turbulence in the compressed gas could be determined. (1) Azr-teghtness of the Compression Space. There were three points at which appreciable leakage was liable to take place :— (a) Round the sides of the main piston. (b) Past the indicator piston. (c) Round the fan spindle. Ignition of Gases by Sudden Compression. 83 Of these, the first two were completely eliminated and the third reduced to something quite negligible. Figures of the actual leakage are given below. The method of eliminating piston leak will be understood from fig. 2. The piston is made in two parts. Below the FIG 2 cast-iron top, which carries a single piston ring of the usual type, there is an aluminium body which for some distance is of smaller diameter than the cylinder. Round this waist are two cup-leathers, C and D, each with the periphery turned upwards, which are separated by a cast-iron ring of square section (H, fig. 2). The whole space above and between the cup-leathers was filled with castor oil so that no air could leak down past the piston head until it had first made a space for itself by forcing castor oil down past the two cup-leathers in series. To eliminate lek past the indicator piston, the plan was hit upon of turning the whole indicator upside down and then pouring in a little castor vil into the space above the piston, as illustrated in fig. 3. Here A represents the piston of the inverted indicator with the space above it filled with G 2 84 Messrs. H. T. Tizard and D. R. Pye on the castor oil. The duct BC in the bracket which carries the indicator is based on the slope to prevent the possibility of any oil getting back along it and into the combustion space at C, This arrangement was very satisfactory, for, besides forming a perfect seal, it served to keep the piston well lubricated ; it was found, moreover, that only a very small quantity of oil was forced down past the piston, even after continued exposure to high pressure. The arrangement of fan and fan-spindle is shown in fig. 4. Here the plug A, shown in section, was made to screw vertically* downwards in the centre of the cylinder PACKING — CYLINDER — head, its lower end finishing flush with the top of the com- pression space. At the highest compression ratios this space is 3 inch deep, so that the piston head under these conditions is close up to the under side of the fan. The fan was driven at varying speed by an electric motor through the usual form of flexible drive used for speedometers. To prevent leakage up round the fan-spindle, this was made long and thin, and was also provided with a cup-leather of which the tightness could be adjusted by the screw B in fig. 4. By using a motor with ample reserve of power for driving the fan, it was possible to keep the cup-leather so tight as to reduce the leakage to a negligible amount. — Ignition of Gases by Sudden Compression. 85 Experiments on Air Leakage from the Cylinder. Observations were made of the rate of fall of pressure beginning 10 seconds after compression had occurred. By this time all pressure-fall due to heat-loss has ceased, and the observed fall represents leakage only. Experiments were made— (a) With fan stationary. Compression ratio 9:1. Contents of cylinder were the products of com- bustion of a benzene and air mixture. se Deflexion on Indicator Time. Screen. 10 secs. after compression and combustion ...... 0°63 inches, 20 r : Satahy Bisse OGL) 3: 30 : - mA We EL Ulsan airs (CRE ee 45 : fe any ae Ain ya oe O56?) 60 B A Stil Bia nes Ooar The indicator calibration was 1 inch deflexion= 188 lb. per sq. in., so that the above gives a rate of fall of 0-1 x 188 50 (6) With fan rotating at full speed, 2000 r.p.m. Com- pression ratio 10:1. Contents of cylinder pure = *38 lb. per sq. in. per sec. hydrogen. Ti Deflexion on Indicator ime. 5 creen. 10 secs. after CoOMpressiON ............seeseseeeserees 0°66 inches. 20 ” oT Oo 0°53 Pe 30 9 59 HAR tee twee eter eeereeres 0°40 a This, owing to the low viscosity of hydrogen and the high compression ratio, was a very severe test. The rate of fall was 26 x 188 20 riment the rate of fall of pressure due to cooling was as high as 600 lb. per sq. inch per second immediately after com- pression, so that the effect of.gas leakage on the apparent rate of cooling was clearly negligible. = 2°45 lb. per sq. in. per sec. Now in this expe- The Indicator. It will be convenient at this point, before going on to describe the timing apparatus, to give some further data as to the indicator used. As stated above, this was of the 86 Messrs. H. T. Tizard and D. R. Pye on the standard Hopkinson optical type *, and need not be described in detail. The piston used throughout the experiments was one of 0:125 sq. in. area. Pressure on the piston deflects a spring attached to it, and thus tilts a small mirror which reflects a point of light from a fixed lamp. The magnitude of the pressure is thus arranged to correspond to the vertical (downward) deflexion of the image of the light on a 44 x 33 photographic plate carried in a camera fixed to the indicator. The calibration of the pressure scale was made in two. ways. Firstly, by subjecting the piston to known oil pressures and measuring deflexions on the camera screen, and, secondly, as a check on the piston area, by direct dead- weight loading of the spring and measurement of the deflexion. The oil pressures in the first calibration were produced by a carefully gauged vertical plunger loaded with weights. The calibration by this method was carried out at the beginning of the experiments, and it was found that the relation between pressure and deflexion was given very closely by a straight line at a slope corresponding to a pressure rise of 188 lb. per sq. in. per inch deflexion. At the conclusion of the experiments this catibration was checked by the dead-weight calibration as follows :— Reading on Camera Screen. Load on Spring. Loading. Unloading. | 0 aes 0°39. in. 0:40 in. Weight carrier = 3°25 Ib. ...... ODA: 0-54 ,, Carrier - 10) = 13°257 ees: 0-975 ,, 0:99 ,, PEO OBI a Men ik 1:40, 1:43 ,, O30 == 8825) 6 a IESE habe 1:81 This also shows a satisfactorily straight-line relationship which checks remarkably well with the previous calibration, for D : 33°2 93-4 |b ead weight per inch deflexion = ao 3 : corresponding to a pressure on a piston of area 0°125 sq. in: O18 x 23:4 = 15722 ib The previous figure of 188 lb. per sq. in. per inch de- flexion has been used throughout the calculations. (2) The Timing Gear. The pressure in the cylinder being given by the vertical motion of a point of light on a photographic. plate, it was necessary for the accurate measurement of the lengths of the delay period to give the point of light at the same time * Hopkinson, Proc. Inst. Mech. Eng. Oct. 1907, p. 863. Ignition of Gases by Sudden Compression. 87 a uniform and exactly known motion across the plate horizontally. To obtain this uniform horizontal velocity, the arrangement in fig. 5 was adopted. A is a vertical cylinder closed at the bottom and carrying at the top a guide for the piston rod B. At the lower end of the rod B is a light loosely fitting piston in which are drilled one or more small holes. The rod B has a collar C above the guide, and can be loaded with weights, up to 50 lb. if need be. The cylinder is filled with paraffin which, when the piston and weights have been raised and then released by a trigger arrangement, is forced at high velocity through the holes in the piston from the under to the upper side. This arrangement gives a velocity of fall of the piston which is uniform to within 1 or 2 per cent., and it was found, moreover, that this velocity varied directly as the square root of the weight carried: a result which shows that the viscosity of the oil is a negligible factor in deter- mining the rate, and that the latter will therefore not be attected by any small changes of temperature which might occur from day to day in the oil. This point was checked experimentally, and it was found that the maximum change in the rate of fall produced by heating the oil from 19° C. to 65° C. was 14 per cent. As the oil temperature during the experiments never varied by more than 2° or 3°, the effects of temperature changes were quite negligible. Piston and weights are suspended by a steel wire from an arrangement of pulleys (F) carried on spindles attached to \ iii 88 Messrs. H. T. Tizard and D. R. Pye on the the cylinder head. D and E are balance weights to keep the wires taut, and F is a compound pulley which reduces the horizontal motion of the wire G to about one-half the vertical movement of the piston. At H the wire G is divided, and each half wraps round the periphery of a sector attached to the arm K of the indicator. The barrel of the indicator is thus uniformly rotated through a sector about a vertical axis, and the speed of the point of light horizontally across the photographic plate in the indicator camera can be varied between wide limits by alteration of the weights and the number of holes in the loaded piston through which the paraffin is forced. Actually it was found in these experiments that only two speeds were required, of 5-77 and 4-30 cm. per second. These were obtained when the piston carried weights of 40 and 24 lb. respectively, and was pierced by a single hole } inch diameter. : Both the speed and its uniformity were measured by in- terrupting the light at. known intervais while it traced a straight horizontal line across the plate. To obtain the uniform interruptions, 60 equally spaced contacts were arranged round the periphery of a disk about 6 inches in diameter. An electric motor was used to rotate the disk through a friction clutch, the speed being kept steady by the operation of a governor, which caused the disengagement of the clutch when the speed tended to increase. The arrangement was very similar to that of a gramophone motor, except that the speed of the latter is kept steady by means of a little brake which is operated by a governor, whereas in our experiments the governor operated the clutch. This apparatus, which was made at the Royal Aircraft Establishment, Farnborough, was lent to us for these experi- ments by kind permission of the Director of Research, Air Ministry. To calibrate the falling weight apparatus, the speed of revolution of the disk was first adjusted roughly to one com- plete revolution in 2 seconds. It was then timed repeatedly over twenty complete revolutions by stop-watch ; the mean time for this number of revolutions was found to be 36:5 seconds, the variations in successive timings not exceeding ‘4 second. Hence, since sixty contacts were made each revolution, the time between the beginnings of successive 36°5 contacts Was 90 = 0:0304 second. The 6-volt lamp of the indicator was then connected up, through the contacts on the disk, to a 12-volt battery; it was thus greatly “overrun” whenever contact was made. The effect of Ignition of Gases by Sudden Compression. 89 overrunning the lamp in this way was to make it flicker brightly, so that when the indicator mirror was rotated by the dropping weight, the reflected light made a horizontal line of dashes across the photographic plate, the interval between the middle of two consecutive dashes corresponding to the interval between two contacts on the uniformly rotating disk. The magnitude and uniformity of the velocity of rotation of the indicator mirror by the falling weight could thus be measured. Records were taken with falling weights of 8, 24, and 40 lb. To show the uniformity of motion during descent, the results of measuring two plates taken with the 24-lb. load are given in full. The apparent variations in any single experiment are due rather to the difficulty of estimating the centre of a flicker to 01 mm. than to any real variation in the rate of fall of the loaded piston. Puate A. Puate B. 5 flickers in 0°70 cm. 0°65 cm. 10 1°34 1:30 15 2:0 1:95 20 2°66 2°6 25 3°30 3°28 30 3°98 3°95 35 4°63 4°58 40 5:29 5:22 45 5°93 585 50 6:59 6°51 55 7:22 Fale, 60 7:90 7°78 65 8:53 8:40 70 9°21 9:06 75 9°89 9-70 Mean 1 flicker in 0:132 em. Mean 1 in 0°129 cm.” 1 flicker = '0304 second. .’. 1 second = 4:34 em. on plate. 1 second = 4°26 cm. Mean 4°30 cm. = 1 second. Similar records with loads of 8 and 40 lb. gave time scales of 2°22 cm. = 1 second and 5°77 em. = 1 second. The means for the two plates with 40-lb. load were 5:75 and 2°79 em. per second, so that it may safely be assumed that the time scale was known correctly to within 2 per cent., and probably less. This is a degree of accuracy as great as that with which, as a rule, it is possible to measure the records. The vertical movement of the weights was about 7 inches, 90 = Messrs. H. T. Tizard and D. R. Pye on the which produced a horizontal movement of the point of light more than twice as great as was necessary to traverse the photographic plate, so that ample time was provided for the speed to have become uniform before a pressure record was taken. The fact that the indicator and all gear except the actual falling weight was carried on the cylinder head, made it possible to raise or lower this when changing the com- pression ratio without affecting the timing gear. The only — difference was an alteration of about an inch in the distance fallen by the weights between the position for maximum and minimum compressions. Figs. 6 and 7 are prints taken from two typical records. Fig. 6. Pressure . lbs. per S?. in. above abpospherce. — — > Jime in Seconds In the first there is no explosion, and the record shows simply the rise of pressure due to compression, and subsequent fall as the compressed mixture cools. Fig. 7 shows the first rise of pressure due to compression, the “‘ delay” period of nearly constant pressure followed by the practically instan- taneous rise on explosion, and finally the rapid cooling of the intensely hot products of combustion. It muy be mentioned that the spring of the indicator was protected from ~ the force of the explosion pressure by a stop which prevented the deflexion from ever being greater than just to the edge of the photographic plate. III. If a gas at a known temperature and pressure is suddenly compressed in a gas-tight cylinder, we can calculate from a measurement of the maximum pressure reached the average temperature of the gas at the moment of maximum Ignition of Gases by Sudden Compression. oy compression. If the volume then remains constant, measure- ment of the rate of fall of pressure with time gives the rate of loss of heat, if the specific heat of the gas is known. One would expect this rate to be closely proportional at any moment to the difference in the average temperature of the fh °o °o w oO ao ny oO {o) . | BAL) Ignition of Heotane and Air \ Maximum compression Termp. 10°C. 100 Pressure, 108, Per $9. 177, 2bove atmospherie pressyre, i¢] 0-25 6-5 0-75 1G Time 17 Secoro’s. gas and that of the cylinder walls. If the temperature of the gas is T° absolute, and that of the walls is 0, then aT — >, =a(T—6), . bein 8y (CE) where “a” is a constant which we call the cooling factor, and which depends on the nature of the gas, and its degree of turbulence in the cylinder. The results given later will be seen to justify this equation. Integration ot (1) gives log. Te 0 OE RS 2 where T is the average temperature when ¢ seconds has ’ 1 elapsed from the moment the average temperature was T). 92 Messrs. H. T. Tizard and D. R. Pye on the Since the total volume remains constant, this equation can be written eer = 09, =e y Jel if the simple gas laws hold. Pyas, is here the maximum compression pressure, P the observed pressure ¢ seconds after the maximum compression pressure is reached, and Py the final pressure of the gas when its temperature is the same as that of the walls. Pyis therefore equal to 7X Pi, where 7 is the’ compression ratio and P, the initial pressure (in these experiments one atmosphere) before compression. Fig. 6 is a typical cooling curve obtained when air at an initial pressure of 14°73 |b. sq. in. and temperature of 23° C. was compressed in the ratio of 7°02 to 1. The time of compression was 0°08 second, and the values of the cooling factor a, obtained from observations of the fall in pressure, are given in the following table :— spat. iw al yee TABLE I, Time after Observed | naan Eye 2°30 max. pressure }, pressure= P | P—P» | log 3p =o. = ae (seconds). lb./sq. in. ice, 0 1907 Fi S6sam aie 185°7 | 82:3 _ 068 ‘90 B47 174°6 taal an ASK ‘87 “H2 | 165 616 194 "86 695 158°3 | 54:9 "244 “81 ‘87 152°4 | 49 "294 ‘78 | Mean=:85 @ (initial temp.) = P,=14'78 lb./ 23°C. Pp=Py x 7'02=103°4. sq. in. Max. temp. (cale.) = 299°C. It was found as a rule that the calculated cooling factor tends to diminish as “‘¢”’ increases. This may be duein part to errors of observation, for the errors are big when the time interval after the attainment of maximum pressure is small ; but experimental error will not wholly account for it, and it may be explained on the reasonable assumption that there is a fair degree of turbulence in the gas just after a sudden compression, which dies down after a short time. Rotation of the fan at a high speed increases the rate of cooling con- siderably ; experiments with air gave a cooling factor about Ignition of Gases by Sudden Compression. 93 three times as great when the fan was running as when the gas was “stagnant,” This will be referred to later when discussing the results of the experiments of ignition. Raising the compression ratio also increases the cooling factor for air. ‘his is also to be expected since the distance between the top of the piston and the head of the cylinder is lessened. The cooling factor for “stagnant” air in the apparatus was found to increase from about 0°76 at a compression ratio of 6 to 1 to about 1:0 at a compression ratio of 10 to 1, the distance between the piston and cylinder head being approxi- mately 4 cm. (1°6 in.) at the lower and 2°3 cm. (‘9 in.) at the higher compression ratio. The experimental error in the cooling factors, obtained by experiments similar to that quoted above, is probably about 5 percent. It should be noted that the rate of loss of heat under the above conditions is considerable. For instance, in the experiment quoted the maximum difference in temperature between the gas and the walls after compression is 276° ; and since the specific heat Cy is about 0°18, the air is losing heat at the maximum temperature at the rate of 0°18 x °85 x 276=42 calories per second per gram. It follows that if an explosive mixture of gases is suddenly compressed to its ignition temperature, in such an apparatus as that described above, the initial rate of the chemical reaction at the lowest temperature at which ignition is ob- served must be considerable, for the evolution of heat due to the reaction must equal approximately the rate at which heat is lost to the walls. For example, the total heat of com- bustion of a mixture of a paraffin hydrocarbon and air, in the correct proportions to burn to CQ, and water, is approximately 700 calories per gram of mixture. If it ignites when suddenly compressed to such a temperature that the rate of loss of heat is 35 calories per gram, the reaction, if it continued uniformly at the initial rate, would be complete in 20 seconds. This illustration may serve to show the general nature of the reactions that occur on sudden compression ; what occurs in practice is that the gas, or part of the gas, reacts so that the evolution of heat takes place at a somewhat higher rate than the loss of heat by conduction etc.; hence the temperature of the reacting gases must automatically increase, and with it the rate of reaction, until the gas ‘‘ explodes.” The interval between the end of compression and the explosion must clearly depend mainly on three factors: (a) the compression tempera- ture, (b) the temperature coefficient of the reaction, (c) the rate of loss of heat to the walls. 94 Messrs. H. T. Tizard and D. R. Pye on the IV. It is now generally recognized that the rate of an ordinary chemical reaction varies with the temperature in a way which may be empirically expressed by an equation of the form B bee TC) ee where k is the velocity constant, and A and B are constants, B being the temperature coefficient. In the case of reactions evolving heat, we can write this : B Q= Ae Ty uc eo aie ee where ( is the initial rate of evolution of heat when a definite mixture is suddenly compressed to the temperature T. At the lowest ignition temperature Ty we have, B Qader where Q) can be measured from observation of the rate of cooling of the gases at a temperature slightly below the ignition point: 2. ¢€ Qo= ae, (Ly 8), ee) where @ is the cooling factor, c, the specific heat of the mixture, and @ the temperature of the walls. Now suppose that the gases are compressed initially to a higher temperature T: the initial rate of loss of heat will then be higher than Qo, namely Qo — 6? but since the effect of temperature on the rate of evolution of heat due to the reaction is so much greater than that on the rate of loss of heat, it is sufficiently accurate at present to assume for our purposes a constant initial rate of loss of heat —(,. The initial rate of reaction at the higher temperature will be given by ae From (5) and (6) we have Qr hs b iy Ty loge =7(2 ae ae where 6=0'4343 B. Now under these circumstances the net rate of gain of heat is (Qr—Q) calories per second. Ignition of Gases by Sudden Compression. 95 The initial rise of temperature is therefore given by aT 4 T 0 = A-W=G(G7—1}- - - Beet) gp a eo or i iE Qo or (o a d (7) =ore Qo But since Qo=2e,(To— 8), we have ] T T,—@ et): a() — a{ dé. ‘ é (10) Qo V. Can this equation be used to determine the delay, or period of slow combustion, that occurs before the temperature suddenly rises very rapidly, i.e. before explosion takes place? Strictly speaking, this could only be done if Q, the rate of evolution of heat during the initial reaction, depended only on temperature and not on the concentration of the reacting substances. According to all ordinary theories of reaction, this would not be true; the rate of reaction should depend in some way on the concentration of the reactants. If we consider such a reaction as the combustion of heptane CHie — 11 = rs CO, + 8 HO, we can hardly suppose it is necessary, before the initial re- action—whatever is its nature—can occur, for 1 molecule of heptane and 11 moleculeg of oxygen to collide; but it is reasonable to assume that the rate of reactiun must at least be proportional either to the concentration of the heptane or to that of the oxygen, or to the product of the two. It is necessary, however, to point out that there is little, if any, satisfactory evidence that homogeneous reactions in gases obey what is ordinarily understood by the law of mass action. In fact, evidence from the ignition of gaseous mixtures by sudden compression points rather to the reverse. Dixon and Crofts’ experiments* on the ignition of mixtures of hydrogen and oxygen are difficult to explain by any reasonable assumption as to the mechanism of the reaction based on the * Dixon & Crofts, Trans. Chem. Soc. 1914, p. 2036. 96 Messrs. H. T. Tizard and D. R. Pye on the law of mass action, even after taking into account the possible effects of a different rate of loss of heat in the mixtures with which they experimented ; the authors state, in fact, that they ‘“‘can offer no satisfactory explanation of the phenomena observed.” Recent experiments by one of us * have shown that the ignition temperatures of hydrocarbon-air mixtures are independent, within the errors of measurement, of the proportion of the hydrocarbon within quite wide limits. For instance, when the proportion of heptane to air was varied in the ratio of 1 to 10, the ignition temperature was only lowered apparently by some 8°C. from 293°C. to 285°C. As the error of observation may certainly amount to 4°C. in such experiments, there is here no evidence that the alteration of the concentration of heptane has any effect on the ignition temperature. If, as might appear reasonable on the law of mass action, the rate of reaction depended directly on the concentration of the heptane, we should expect it to be 10 times as great in one case ag in the other. Now, it is shown later in this paper that the temperature coefficient B, in the case of heptane, is about 13,000, from which it follows from equation (5) or (8) that if the initial rate of reaction on com- pression were really 10 times as great in one case as in the other, the ignition temperature of the richer mixture should be over 50° lower than that of the weaker mixture. One could not fail to detect with certainty a difference of this order: hence from the experimental results we come to the conclusion that the rate of combustion under these conditions does not depend on the concentration of the heptane vapour within wide limits. Experiments with other similar sub- stances support this view. VI. If this be correct, the only probable alternative is that the rate depends essentially on the concentration of the oxygen. We have not yet attempted to put this to a direct _ test in the apparatus used by us, since it would be necessary, to do this completely, to explode detonating mixtures of hydrocarbons and oxygen; and we were anxious to avoid the danger of breaking the apparatus before other important experiments were carried out. We intend, however, to examine this point in the near future. It remains to consider whether, if the rate of reaction is directly proportional to the concentration of oxygen, the effect of the automatic decrease in oxygen content on ignition can be safely left out of account in calculating “delays” by * Tizard (oe. cit.).° Iynition of Gases by Sudden Compression. 97 means of equation (10). For this purpose we have to estimate to what extent the reaction occurs during the slow period ot combustion before the explosion occurs, and the pressure- time curve becomes almost perpendicular to the time axis (see fig. 7). Taking the experiments on heptane as an example, we find (Table III.) that the biggest delay observed was about 0°6 second ; that the ignition temperature under these conditions is 250° C.=553 absolute, and that the differ- ence of temperature between the gas and the walls was about 240°C. The observed cooling factor was 0°51. Since the specific heat of the mixture experimented with is approxi- mately =0°2, the rate of reaction at the ignition temperature must correspond to a heat evolution of 0:51 x 0°2 x 240 = 24 calories per gram per second. The total heat of combustion of 1 gram of the mixture (containing about 75 per cent. of the theoretical amount of heptane for complete combustion) is about 510 calories, so that it is evident that during the period when the temperature is only rising slowly, which is always less than half a second, the amount of reaction and therefore the changes in con- centration must be small. Once the temperature begins to rise quickly, it is evident that the disappearance of oxygen can have only a secondary effect on the rate of the reaction compared with that of the rise of temperature until the com- bustion is nearly complete, so that the error involved in the calculation from equation (10) of the total time of combus- tion, by ignoring the effect of changing concentration, must be small. In view of the unavoidable experimental errors in carrying out experiments of this kind and of our still in- complete knowledge of the mechanism of combustion, we do not, in fact, think that any attempt to take fuliy into account such secondary effects is Justified at present. VII. We therefore arrive at the conclusion that the time for complete combustion of an explosive mixture of gases when suddenly compressed to a temperature above its ignition temperaiure is closely given by the integration of equation (10), and is therefore l we 1 ( ty Se ———d | ar My yi 1 naa & (i) (11) a ( ak To T/T Qo where T, is the lowest ignition temperature under the con- ditions of the experiment, and T is the initial compression temperature. Phil. May. Ser. 6. Vol. 44. No. 259. July 1922. H Wn 98 Messrs. H. T. Tizard and D. R. Pye on the In this equation, e is given by equation (8); its value , . depends on the magnitude of the temperature coefficient of the reaction. It is not possible to integrate the equation completely, but the integration can be carried out by approxi- mate methods for any value of T° In the table below are 0 penis “ given values of the expression 2 (ee nN at different values of —,. and for various values of —. The ae a, corresponding curves are shown in fig. 8. o[To-6]t ; To 0-1 G-2 0:3 Tf our views are correct, these should form standard curves representing the delays which should be observed under different conditions when any explosive mixture of gases is compressed to a temperature above its lowest ignition temperature. ‘The application of the theory to any specific case should enable the temperature coefficient of the reaction to be determined. Ignition of Gases by Sudden Compression. 99 TABLE II. T)—@ age T l Values of «a(~° “ji for different values of = and , ross a5 Lo 1 T/T,. | o/T,=12.| =10. | =8 | =6. =4, 1004 | -086 Oe gee beg ae. VOL ‘056 "OTD "108 ‘169 321 1:03 "0245 036 ‘056 ‘098 *209 L-OF ‘OOS1 O42 4.) “O25 onset eke rPk ‘0033 ‘0065 | Ola, sue ‘095 1-27 “0002 ‘0007 | 0022 ‘0083 | ‘0415 The curves bring out clearly the effect of the two main factors which determine the characteristics of an explosion by sudden compression ; namely, the initial rate of loss of heat, and the temperature coefficient. If two gases have the same lgnition temperature under the same conditions of loss of heat, the sharpest explosion will occur in the case of the gas with the highest.temperature coefficient, and the greater in this case will be the effect, on the magnitude of the delay before explosion, of a higher temperature of compression. On the other hand, in any one case, the ignition temperature will be raised by carrying out the experiment under con- ditions which involve an increased rate of loss of heat; at the same time the sharpness of the explosion will also be in- creased, VIII. To test the above theory, and to use it to obtain a measure of the temperature coefficient in certain cases, we chose three substances: heptane C;Hj,, ether C,H;.0.C,H;, and carbon bisulphide CS,. These substances were chosen for the following reasons: (a) they could be obtained in a sufficiently pure state; (b) they all have low ignition tem- peratures, which lessens the practical difficulties of the experiments ; (c) they are known to behave very differently from the point of view of detonation when used as fuels for internal combustion engines; and (d) their difference in chemical and physical properties makes the comparison of their behaviour on combustion particularly interesting. To test the theory adequately we considered it absolutely necessary, particularly in view of the simplifying assump- tions made, not to be content with one set of conditions for the ignition experiments. Two series of experiments were therefore made with each substance; in the first series the H 2 100 Messrs. H. T. Tizard and D. R. Pye on the gaseous mixtures with air were compressed in a non-turbulent condition, while in the second a high-speed fan was kept running throughout the period of compression and subsequent ignition. The use of the fan increased the rate of heat-loss at the compression temperature by about three times ; hence the difference in ignition temperatures observed with and without the fan running gives an important and necessary check on the value of the temperature coefficient which is calculated from the “ delay”? curve obtained when the non- turbulent gases are compressed. The temperatures given below represent the average temperature of the gas at the instant of maximum compression. By measuring the com- pression pressures in each experiment, a value of “vy,” the apparent ratio of the specific heats, is obtained from the expression : aes aE pt le 2) where 7 is the compression ratio. The average value of y is taken for the series, and the compression temperatures then calculated for each case from the expression Ae Ag! ipl —! @ e In each set of experiments the initial mixture of gases was of the same composition throughout, the proportion of air being somewhat greater than that required for complete combustion. The intention of using a weak mixture was to avoid as far as possible the deposition of carbon ; as stated above, the absolute ignition temperature is not affected appreciably by fairly wide changes in the original strength of the mixture. IX. The first results with heptane gave a very satisfactory confirmation of the theory developed above. The results of the experiments are given in the following tables and diagrams, which include measurements from all the records made under each set of conditions. No unsatisfactory records have been discarded. Ignition of Gases by Sudden Compression. 101 TaBueE III. Ignition of mixtures of heptane and air. Mean apparent value of “y” observed = 1°313. Fan stationary. Initial pressure (atmospheric) = 14°8 Ib./ sq. in. in expts. A; to Ayo, and 14:9 |b./sq. in. in expts. Ay, to Aj;. Strength of mixture = 1 grm. heptane: 20 grms. air. | Max. Avge. temp. Delay Cooling No. of | Compn. | Initial Max. Compn. See ponte expt. ratio. | temp. pressure. y=1°313. obs. Ley ! eee Boo | | 51°C. 405 1b./sq. in. | 280° C. No ignition. 0:49 ee 6:03 50°5 ~~ | 156 295 0:19 sec. _ aS 702 | 505 192 323 0-04 = Bape] 802 | 49°5 225 346 0-007 — Perec Oo. .| 46 | 141 273 No ignition. 0°53 Benes. | 582 46 | 151°5 281 0°56 _ : PEERS | 623 | 46 163 293 0:21 _~ Oe 802 | 465 227 340 Very small. ). ARS 7:02 4 189'5 316 0-06 aay fascees 702 =|. 42 «166 307 0:07 —— :. Sanne | 656 58 174 324 0:05 — ORE es 6°56 53 Dae 315 0:06 _ 1. es | 6°56 48 176 306 0-12 — aa exteie - 6°56 44 pales 298 0-18 — |: See 6°56 41 | 176 | 293 0°28 — , eee | 6°56 38 178 288 0:25 = Ce 6°56 39 176 282 | 0°58 — The cooling factors in Table III. were obtained from the results of those experiments where no ignition occurred by the application of equation (3). The results for A, were as follows :— TaBLeE LV. Calculation of cooling factor. wa i 3 | e | Time from Obs. pressure. | Lr lays 0. ‘max. pressure.| 1b./sq. in. Pde | log P—Pr = 0. ae 2°35, 0 140°5 58°5 aa — 0°23 sec. 134 a2 0512 | 0-51 0:47 127°5 45°5 | "109 | 0°53 0-7 | TD i saa | 149 | 0:49 0:93 119 toe 199 | 0°49 1°16 116 | 34 ‘236 | 0°47 1°39 | 112°5 30°5 "283 0°47 Mean=0'49 P= 5°50 x 148 = 821 lb./sq. in. 102 Messrs. H. T. Tizard and D. R. Pye on the The results for plate A; were similar, the mean being a = 0°53. In the calculations the figure 0°51 has been taken. TABLE V. Tgnition of mixtures of heptane and air. [an full speed. Initial pressure = 14°86 lb./sq. in. Strength of mixture as before. Mean a value of y = 1°310. No. of | Compn. | Initial |Max. Compn.) Max. Aver. Cooling | | expt. | ratio. temp. | are | temp. cale. Delay ees | Coning | B, 6:03 \G) he OF 157°5 310° C. 0°16 sec. —- B, 5:52 59 139 291 No ignition. | 1:43 | Bs D738 595 148 298 i 1-40 | B, 5°91 595 154 304 4 1°36 B; 6:13 59-5 161 310 ob 1-47 | B, 6°33 59'5 165 313 0°13 = 1s 6°56 o9 175 321 0:08 — B, 701 60 188 336 0:05 — ndBy, 8:02 60 e 362 tO: 00 — Mean= pose * Tonited before top of compression. X. Considering firstly the results of the experiments without the fan running, we find that at the lowest ignition temperature of about 280° C., the ae of the walls Gye) was about 40° C. The expression « an ”) (equate ahale) | Lo has therefore the value— 0-51 x 240 D938 Assume that the true ignition temperature under these conditions is 280° C. = 553 absolute, and that ~ =10: 0 then by integration of the theoretical equation (see Table II.) we obtain the following results :— = Pa, Ignition of Gases by Sudden Compression. 103 TasBieE VI. Ignition of heptane by air. No fan. Lowest ignition temperature Ty taken as 553 absolute. 7 assumed = 10. 0 iy lh ie 1° QC, a (eae ) ¢ (theor.). t(calc.). | 0 1°004 555 282 0-112 0°51 sec. | 1:01 558°5 285°5 0-075 OBE as >| 1:08 569°5 296°5 0-036 O1G4... | ..| 1:07 592 319 0-014 OuGsi | ues Ga 614 341 00065 Ce 0 1:27 702 429 0:0007 0-003 | J The last column contains the theoretical ‘“ delays” that should occur at compression temperatures given in the third column, if the temperature coefficient of the reaction L 5 corresponds to a value of ~ = 10. 0 Fig..9 400 ~) G Heptane and Aji, ae! & Y 350 No ignition. 2 03 0: Seconds delay The corresponding theoretical curve is shown in fig. 9, the experimental points taken from ‘lable III. being marked with a cross. The general agreement is all that could be desired. No ignition. 4 0 O°! O- 5 of a ke 104 Messrs. H. T. Tizard and D. R. Pye on the XI. If the value for the temperature coefficient so deduced is correct, it should be possible to use it to calculate the higher ignition temperature when the fan is used, and also the shape of the new delay curve. Now, the mean cooling factor with the fan has been shown to be 1°42. The ignition temperature under these conditions is evidently about 310° C. (see Table V.), the temperature of the walls being 60° C. Hence the ratio of the rate of loss of heat with and without fan at the respective ignition temperatures is— Qo 142 (810—60) Q) 0°51 (280= 40) Hence the new (theoretical) ignition temperature Ty’ (with fan) should be elven by (see equation (8))— b Ae log Q- = log 290 — Te = a = PU. ise. 0-462 = 10 (1 — Ty To Diao a or Le = 0°954 i = Sol) c= B07? Ok This is close to the observed value. bi ail The new value of Ad will be eek Te = 9:5; while the value 0 Ti a) _ 1:42 x 247 580 The corresponding theoretical values of the delay are shown in the following table. TABLE VII. Ignition of heptane by air. Fan full speed. Ignition temperature calculated from previous results, 580°C. = 0°60. b ==9° ae | y , a: nya .. 10 ml ==") Haheor)o| tesla \ ‘Lp | 10045) Wy yr582 349 0:12 | 0:20 see, 100 |) 4 586 313 | 0:08 a Ons 105 (M0 597 324 0-04 007 Oy GL 348 0-016 0-027 LI | 644 371 00076 0-013 Ignition of Gases by Sudden Compression. 105 The theoretical values given in this table are represented by the dotted curve in fig. 9, the experimental results (Table V.) being shown by circles. The close agreement between experiment and theory is obvious; it is, indeed, closer than could reasonably have been expected in view of the fact that the temperature errors must be estimated as 3 or 4 degrees, while the cooling factors are subject to an error of about 5 per cent. The results can, however, leave little doubt of the substantial accuracy of the simple theory worked out above, and the temperature coefficient deduced must be very near the truth. It is of great interest to note that it is of the same order as that of chemical reactions in liquids at ordinary temperature ; for the reaction velocity is approximately doubled for a 3 per cent. rise in absolute temperature. XII. The only experimental values given in Tables III. and V. which seem to call for any special comment are those corresponding to experiments A, and A,g. The ‘delays’ found in these experiments were considerably smaller than those expected theoretically. This may be due to the fact that the measurement of very small delays is necessarily somewhat inaccurate with the apparatus used, since the speed of the piston falls off as the compression approaches its maximum. In such cases the lowest ignition temperature is, of course, reached before the piston reaches the top of compression, so that the measured ‘‘ delays ” which are measured from the time of maximum compression tend to be too small. But there is also a curious effect, which is invariably observed in these experiments on the self-ignition of carbon compounds, when the initial tem- perature is high, and the time of explosion short. It is always found that the explosion, though apparently sharp, is not complete, but that a fluffy deposit of carbon is thrown down. This deposition of carbon in an explosion has often been noticed by other workers when ignition is effected by a spark, but it is usually thought to be a consequence of having too little oxygen for complete combustion; in our experiments, however, the oxygen was always in considerable excess. When the minimum ignition temperature is not greatly exceeded, and when therefore the explosion is comparatively slow, combustion is complete, and no carbon deposit is formed. At higher initial temperature, however, one cannot escape the conclusion that the hydrogen is burnt preferentially to the carbon, and that the rate of combination of carbon atoms can be greater than the rate of combination 106 Messrs. H. T. Tizard and D. R. Pye on the of carbon with oxygen. The exact conditions when this occurs seems well worthy of further investigation. It is always necessary to clean out the cylinder carefully after such a deposit has been formed, and before the next experiment is made; for, if not, abnormal results will be obtained, and the “delay” before ignition occurs will be very much shorter than is expected. To explain this it does not appear to be necessary to attribute any special “catalytic” activity to the carbon; a simple physical ex- planation seems to be sufficient. Such a deposit is known to be a very bad conductor of heat. If left on the walls of the piston and cylinder, we shall therefore have, on the next compression, large portions of gas from which the beat cannot get away quickly. Hence the ignition temperature is lowered, and the explosion takes place more rapidly. XIII. Experiments on the self-ignition of mixtures of the vapour of ethyl ether C,H;.O.C,H; gave very similar results. The results of the experiments on the compression of non-turbulent mixtures are shown in Table VIII. ; while AW Nssicoe WOEUE. Ignition of mixtures of ether and air. Fan stationary. Initial pressure (atmospher ic) = 14°77 lb./sq. in. Mean apparent value of “y” = 1°309. Strength of mixture = 1 Le ether to 15 of air by wee No. of coe. Initial | Max. Compn. ee oe nee Cooking expt. | ratio. temp. pressure, y= 1309. s Ens Ci | ol 25°C} 105 1b./sq. in. 201°C. | No. ign.| 0°47 C, 4°83 24. 116 4 211 Ms 0-47 | OF 5:02 23 APS) as 214 O41 | — | Cy 5:21 23 128 bs 220 0:30 — C,. 5:42 (23 BORD Sy, 226. >| 10-205 — Or; 5°63 23 143 sp 232 N12 — CP 5°82 23 149 :; 237 0093 | — C, 6:03 23 155 4 242°5 0:071 “= C, 656 | 238 172 43 256 0 035 — Table X. shows the results when the fan was rotating at full speed. No ignition was observed in experiments C, and ©, in the first series, and in experiments D,, D., and D, in the second series. The details of the calculation of the cooling factor corresponding to experiment C, are shown in Table [X. The cooling curve was not so regular in the case of ©,, but the mean value of the constant was the same, Ignition of Gases by Sudden Compression. TABLE LX. Jaleulation of cooling factor corresponding to C. 10 Time from | maximum en Pe log pressure. — 7 0:0 116:2 449 0232 sec ELST 40'6 “A GD), 6: 107'3 36:0 ...| 802 | 48 224 = 346 0:03 — | ce ee ie ec We ane > ? ‘ 391 [Ignit’d be 9 tore top of j compn. Mean = 1°34 The cooling curves were very uniform in the above cases. 112 Messrs. H. T. Tizard and D, R. Pye on the XVI. It is clear, from the shape of the curve connect- ing time of ignition with temperature, from the results of experiments without the fan, and from the difference in ignition temperature observed with and without the fan, that the temperature coefficient of the reaction is low. If we take Ty, the ignition temperature without the fan, to bem eiovO.= 526 ‘absaltandan = To theoretical values for the time of ignition from equation (11) which are given in Table XVI. = 7, we obtain the TSO 253 —32 The value of a—°,~ is 0°46 X —___— =0:193. Te 526 TaBLE XVI. T,=526 abs. }/T,=7-0. T,—0 T/T,. | T(abs.). | °C. a( T. Ve (theor.). ¢ (cale.), 1004 | 5228 255 0-187 98 sec 1:01 531 258 0133 704 1:03 5423 269 0-072 38) 1s 1:07 563 290 0 036 RG shee tala 584 oll 0:020 (Gar NLA 66S 395 0-004 021 ,, Fig. 11 at and aif. . 350 ma S [IN B. Time scale half that jof Figs. |IO and it.J | S R 300b- = = No ianitlion. 4 Gea ‘aan = aes O-1 02 03 04 0-5 0-6 0:7 0:8 0-9 Seconds delay, The theoretical values are shown by the lower curve in foe A, ao Lgnition of Gases by Sudden Compression. 113 The cooling factor when the fan is used is 1°34. The ratio of the loss of heat (or reaction velocity) at the ignition temperatures with and without fan is consequently Qo __ 1:34 (296 —42) Qo 0°46 (253 —32) =O: Hence the new ignition temperature T,' should be given by the expression log 3°35=0°525 =6b/T» (1 - = 0 Tye =7(1- 7?) Shire yeaa or T,’ =1—0:075 =e Ty’ =568°5 = 2955 OC, The new value of <”- corresponding to this is therefore To’ b 526 mate ( 555) =6°5. The calculation of the corresponding delay curve is shown Ui a aa in Table XVII, tho value of «(= *) being 0 204 1°34 x 569 == 06: TaBLeE XVII. | | ° a (=| t (theor.). 1 Li ea iy t°C, T,! t (eale.). | 1008 | 571 | 298 0-205 34 sec._| Ot 574 301 0°15 rt 1:03 585°5 312°5 0:08 Sys | 1:07 6085 | 335°5 0:042 OF cl Ill 631 | 3858 0:025 gic sewer | This is the dotted curve shown in fig. 11, the experimental points being marked as before. The agreement between Phil Mag. 8. 6. Vol. 44. No. 259. July 1922. I 114 Messrs. H. T. Tizard and D. R. Pye on the theory and experiment is in this case only moderate. In particular, it will be noticed that although the ignition temperature with the fan running is calculated, from the results without the fan, to be 295°°5 C., actually no ignition was observed at 300°C., and even in one experiment at 307°C., although this is extremely doubtful, since in two other experiments ignition was observed to take place at 307°C. and 304° C., with the comparatively short delays of 0°11 and 0:14 second respectively. It is possible that when the loss of heat is considerable, and the temperature coefficient small, there is an appreciable error introduced in neglecting changes of concentration when calculating the time of ignition at temperatures near the ignition temperature. This would account for no ignition being observed when a long delay was expected. The value of 6/T,) given above cannot, how- ever, be very far wrong. For suppose we take the ignition temperature Ty’ of the mixture when the fan is running as 300° C. instead of 295°°5, and calculate the temperature co- efficient solely from the difference in Tp and To without regard to the “delay” curve when the fan is stationary ; then we shall have Tpe526 hs D7 = 573 70918 and 20, = 1:34 (300-42) Lae Qn.) 0162538 232) ek ee Ne ticg log 3:40 ="531 = a (1— ry b mp SUE e =6°'5 instead of 7:0. Lo This value for < would, in fact, fit the lower part of the 0 delay curve without the fan rather better than the value 7-0, but the calculated delay curve when the fan is running would then be some way from the experimental points. If we take a mean value 6) T= 6-7, we shall be very unlikely to be as much as 10 per cent. from Ignition of Gases by Sudden Compression. 115 the true value, even when all possible sources of error are taken fully into account. XVII. It has already been mentioned that when the time of ignition by compression of hydrocarbons (and of ether) is small, 2. e. when the gases are suddenly compressed to a temperature well above the ignition temperature, carbon is thrown down, even though excess of oxygen is present. In the experiments with CS, an even more curious phenomenon was noticed. In this case, whereas the sulphur burns to SO, when the initial temperature of compression does not exceed very greatly the lowest ignition temperature, when the initial temperature is high the products of combustion smell strongly of H,S. For instance, the products of com- bustion in experiments E, and EH,, above both smelt strongly of HS, although in H,—Ej) inclusive only SO, could be detected by smell. It was also possible to detect H,S after experiments F, and Fy) (with the fan), the smell being particularly strong in the case of Fy). The smell of H,S could also just be detected along with SO, in experiment Fy, whereas in the remainder only SO, could be detected. The H.S could only have come by combination with water- vapour present in the air, which was not dried. This occurrence of H.S is all the more interesting since it is known that a perfectly dry mixture of CS, and oxygen can be exploded by a spark, whereas perfectly dry mixtures of other gases, e. g. carbon monoxide with oxygen, cannot. It is possible that some such reaction as CS, + 2H,0=C0,+2H,8 takes place, followed by the combustion of H.S ; but even if this is the case, it would be expected that the H,S would be quickly burnt in presence of excess of oxygen under the conditions of these experiments. Further experiments on the ignition of H,S itself will probably throw some light on these observations. XVIII. In Table XVIII. are summarized the chief results of the experiments described above. The ignition tempera- tures represent the lowest average temperatures at which the non-turbulent mixture could be caused to ignite. The rates of evolution of heat at these temperatures for the three cases are calculated from the cooling factors and the specific heats of the mixtures. i? 116 «=«©Messrs. H: T. Tizard and D. R. Pye on the TasLe XVIII. | Carton Heptane Ether : ESCs: ol) CEO de, ie Composition of gas by weight... 1: 20 of air. i: 15 Lis | ee | T,=ignition temperature ...... ee 200. a DAC 253° ©. Cy at ignition temperature ...... e020 | 0:20 0-18 | | Rate of evolution of heat due to, 25 calories | : i | | 18 18:5 reaction per gram of mixture per second. ERO IU TIES OE EDL DDE eae calories. | 510 calories. | 386 calories. gram Ol MIXVURC © yee sass cen | | ASL Meee A Sat ENO Bele x zs] pa aeEY Waltie of 6/Ese £2). AS o4 100+5°/, | 110+5°%, | 67+10°/, | | XIX. In order to calculate the true temperature co- efficient B (see equation 5) from the values of 6/To, it is necessary to examine the significance of Ty a little more closely. As already stated, Ty is a measure of the lowest average temperature of the gas at which ignition takes place. Now the actual temperature of the gas after sudden compression can hardly be uniform throughout ; in fact, when the gas ignites after a considerable delay, it is always found that the pressure, and therefore the average temper- ature, falls, in some cases quite considerably, before ignition takes place throughout the mass. This shows clearly that that portion of the gas which ignites at first has initially a higher temperature than the average, thus confirming Dixon’s experiments. Absence of information as to the temperature gradients which may exist under these conditions has no doubt led Nernst and Dixon in their experiments to calculate the ignition temperature as if the compression were adiabatic, and to ignore the influence of loss of heat during compression and before ignition. They assume, in fact, that that portion of the gas which does ignite is at the adiabatic temperature. It is hardly likely, however, that big differences in temperature exist after compression when the gases are in a turbulent state ; and the fact that the temperature coefficients, calculated from the differences in “average” ignition temperature between turbulent and non-turbulent mixture, Ignition of Gases by Sudden Compression. Ly agree so well with those calculated from measurements of time of ignition at various temperatures with non-turbulent mixture, confirms the views taken in the previous paper (Tizard, loc. cit.) that it is unlikely also that any big differences of temperature exist in the non-turbulent mixture after compression. In the absence of direct evidence on this point, however, it is important to calculate the ‘ adia- batic ” temperatures also in the above cases. The mean specific heats C, per gram molecule are : Heptane (room temperature-300°C.) =50 calories approx. Ether (65°-230° C.)=33°6 calories (Regnault). CS, (70°-194° C.) =10°0 (Regnault). Taking these values, and C,=5-0 for air, we obtain the fioures for the mean true value of “ry,” and the corresponding adiabatic temperatures, given in the following table. TABLE XIX. Heptane. | Ether. CS... } | Mean apparent value of | rye TODSELVEU) si... 1°313 | 1-309 1°332 Mean true value of y ... 1353 | 1:347 1:384 “« Average” ignition tem- | 2 2 a a eee 553 abs. 485 abs. 526 abs. | ‘“* Adiabatic” ignition | | temperature ............ 594 516 572 PUI EROMCEY rods. ceca se ANE C4 Sof ole ©. 6749), | 46°. 0.==8:°7.°/,, | | The average specific heat for CS, taken in the above calculations is probably too low, since it refers only to a range of temperature up to 194° C., whereas the ignition temperature was 250° C. | It will be observed that the difference between the average observed and the theoretical adiabatic temperatures is not very great ; we consider that the “average”? temperature is probably closer to the true ignition temperature than is the “adiabatic” temperature, but for the purpose of estimating every possible source of error in the temperature coefficients, it is better at this stage to recognize the uncertainty, and take for the true values of the ignition temperatures the values Heptane 573° absolute thers = 500?" +4 per cent. CS, 549° 39 118 Messrs. H. T. Tizard and D. R. Pye on the Hence, from the values of = we get finally for the true temperature coefficient B : p Temperature coefficient B (equation 5). Heptane-air... 13,200+ 9 per cent. Hiher- pk 20 oule2 eG (0c eo ae CS.- fp 4 COW ORE ees The significance of these figures will perhaps be better appreciated by the statement, that the percentage rise in absolute temperature necessary to treble the reaction velocity is 4 per cent. in the case of heptane and ether and 7 per cent. in the case of CS». XX. Of recent years, considerable attention has been directed to the “radiation” theory of chemical reactions. According to this theory, the ultimate cause of any chemical reaction is to be found in the absorption of radiation of a frequency which depends upon the nature of the reactants. In the case of the majority of chemical reactions, namely those which are not “‘ photochemical ”’ in nature, this radiation will belong either to the visible, or more usually in the short infra-red part of the spectrum. The supporters of the theory hold the view that it is only through the absorption of such radiation that a molecule is able to acquire that excess of energy, over the average at any temperature, which enables it to decompose or to react with another molecule. The frequency of the radiation is therefore known as_ the ‘activating ”’ frequency. This reasoning leads to the conclusion that the temperature coefficient B of a mono-molecular reaction is determined by the relation hy where “v” is the activating frequency, which should correspond to an absorption band in the reacting species. No reliable experimental evidence has yet been brought forward in support of this theory, but in view of the scanti- ness of the data existing on homogeneous gas reactions, it is of particular interest to apply it to the results of the experiments described above. In attempting to apply the theory, a difficulty at once arises. Hvidence has been brought forward to show that the ignition temperature of substances with oxygen is Ignition of Gases by Sudden-Compression. FY practically independent of the concentration of the com- bustible substance. If the rate of the reaction were deter- mined solely by the amount of oxygen present, we might expect the temperature coefficient also to depend solely on the oxygen, and therefore to be the same in all cases. This is clearly not true. Nor does the temperature coefficient, in the case of the heptane explosion which has been most closely investigated, correspond at all closely to that calculated by means of equation (12) from the infra-red absorption of oxygen. Oxygen has an absorption band corresponding to N=3'2 mw, or v="94x10"; hence, since 7 = 486 LO. we should have B=4550 (cale.) instead of 13,200+9 per cent. (obs.). It is clear that equation (12) cannot be applied. On the other hand, if the rate of reaction depended on the product of the number of active molecules, both of oxygen and the other reactant, we should expect, on the same theory, to find the temperature coefficient given by / B= + (+), where v,, v2 correspond to absorption bands in the reacting substances. Now, all hydrocarbons have a weak absorption band at A=2'4 w, and a fairly strong one at 3:43 pw. Taking X=2°4 uw, which is most favourable to the theory, we have v= 1°25 x 10" and vy, (oxygen) ='94 x 10. Hence B= *-s600.10 2-2 10 = 10; 700: This approaches more closely the experimental value B=13,200+9 per cent. It must be pointed out, however, that this approximate agreement is only obtained by an assumption as to the actual mechanism of the reaction which does not agree with the existing experimental results. The failure of the “radiation theory” to account for the results obtained in these experiments is more significant when we regard it in a different way. The theory requires that the rate of a chemical reaction should be proportional to the density in the reacting system of the radiation which is absorbed by the reacting substances. Now, in the case of gases which are caused to react by a rise in temperature due to sudden compression, the radiation density must remain practically unchanged, for the temperature of the walls remains constant. It may be momentarily increased 120 Messrs. H. T. Tizard and D. R. Pye on the owing to the sudden compression, but such an increase cannot persist during the period of delay, and in any case is negligible compared with the increase in radiation density which would occur if the temperature of the walls of the vessel were raised to the compression temperature of the gas. Again, the emission and absorption of radiation by the gas itself at the compression temperature of 500-600 absolute is negligible compared with that of the solid walls. Hence we arrive at the conclusion that, although the density of radiation in the system is not appreciably changed, the gases react ata high rate. This fact appears to us to prove conclusively that the radiation theory cannot be accepted either in its original form, or as modified to meet its failure to account quantitatively for the temperature coefficients of chemical reactions in liquids under steady conditions of temperature. It must be pointed out, however, that in spite of this con- clusion, there does seem io be an indirect connexion between the effect of temperature on the rate of combustion of many substances and their absorption of infra-red radiation. For example, Coblentz has shown that all paraffin hydrocarbons have very similar absorption spectra, with a weak band at A=about 2°4 uw, and strong bands at X= 3°43, 6°86 yp, etc. Now, Ricardo hag shown that the tendency of hydrocarbon fuels to detonate in an internal combustion engine depends consistently on their ignition temperatures as determined in the manner described above. According to our views this is strong evidence that the temperature coefficients are practically the same throughout. Again, it has been shown that ethyl ether has approximately the same temperature coefficient of combustion as that of heptane; while Coblentz has found that its absorption spectrum is also nearly identical, with bands at 2°4 wand 3°45». Carbon bisulphide, on the other hand, has a much smaller tendency to detonate in an internal combustion engine than heptane, although it has a lower ignition temperature; corresponding to this we find that the temperature coefficient is low, and that the first strong absorption band in the infra-red occurs at 7X=4°6 p. Finally, hydrogen “detonates” easily in spite of its high ignition temperature ; its temperature coefficient must therefore also be high, a deduction which is confirmed by some preliminary experiments we have made on the delay before the ignition of a non-turbulent mixture of hydrogen and air, We should expect from this point of view to find an absorption band in the short infra-red region (say about 1:0 w); actually no absorption is observed, but that the Ignition of Gases by Sudden Compression. 121 frequency of atomic oscillation is high is in agreement with our general knowledge of the hydrogen molecule. In spite, therefore, of the strong arguments that have been brought forward against the radiation theory of chemical reactions, these results support the view that there is a connexion, even though an indirect one, between the temperature co- efficients of gaseous reactions and the infra-red spectra of the reacting substances. XXI. The results of this investigation may be summarized as follows :— (a) Quantitative experiments confirm the view that at the lowest ignition temperature the heat evolved by the combustion of a gas just exceeds that lost to the surroundings. (>) From measurements of the rate of loss of heat just below the ignition temperature, and of the intervals between the end of compression and the occurrence of ignition at different temperatures, it is possible to deduce the temperature coefficient of the gaseous reaction. (c) The temperature coefficients so obtained are confirmed by the increase in the minimum ignition temperature which is observed when the gas is in a turbulent | state. | (d) The results show that the temperature coefficient of the combustion of carbon bisulphide is much lower than that of heptane or ether. This is in agreement with the relative tendencies of these fuels to detonate in an internal combustion engine. (ec) The results do not agree with the radiation theory of chemical reaction. (7) Some evidence is put forward to show that the rate of reaction on sudden compression is independent within wide limits of the concentration of the com- bustible gas, but only depends on the amount of oxygen present. ‘This evidence is, however, incom- plete. Weare greatly indebted to Messrs. Ricardo & Co. for the loan of their apparatus and for much additionil assistance ; also to the Department of Scientific and Industrial Research for a grant towards the expenses of the investigation. We also take this opportunity of thanking Mr. C. ‘I. Travers for his help in carrying out some of the experiments. an sss ——_— a SSS SS SS eee ee SSS SS SSS ae [#10859 IX. On the Vibration and Critical Speeds of Rotors. By C. Rongers, 0.6:2., B.Sc., B.iing., ML b ee ING eee US papers have been written on the question of the whirling and vibration of loaded shafts and kindred subjects, and the calculation of the first critical speed—the lowest speed at which the vibration shows a maximum value, is now a matter of daily routine in designing offices. This critical speed can be calculated with sufficient accuracy for practical purposes and as a rule the running at speeds not in the neighbourhood of that indicated by the calculation is free from vibration. But cases occasionally arise where troubles from vibration occur at speeds above or below the calculated critical speed, the reason for which is obscure and the remedy correspondingly difficult to find. It is the object of this paper to discuss various subsidiary causes which might conceivably lead to unsatisfactory run- ning at other than the usual calculated critical speed, but while these are indicated as possible causes of disturbance, it is not to be assumed that these causes always exist or that they will always induce disturbed running. The object is rather to indicate reasons why vibration might possibly arise and thus if an actual case occurs, to suggest a clue to the cause. Although the basis of the paper is a physical or mechanical one, the treatment is largely mathematical, as it is only by this means that formule can be obtained from which numerical results can be worked out. The phenomena when a rotor vibrates are complicated, as the shaft is supported in the bearings on a film of oil, the thickness of which is continually changing, the bearings and foundations are not themselves perfectly rigid, and there is a certain amount of initial bending of the shaft (and toa much smaller extent of the rotor body) due to gravity. If the rotor consists of a number of disks as in a steam turbine, there is also the inter-action of the forces of each disk on the others. For the sake of simplicity, we shall confine our attention to the case of a single part rotor, either a disk or a cylinder, rigid as regards bending and mounted on an elastic shaft running in rigid bearings. Some effects of non-rigidity of the rotor and bearings and of alterations in the thickness of the oil film in the bearings will be indicated. * Communicated by the Author. On the Vibration and Critical Speeds of Rotors. 123 A single part rotor can vibrate in either of two ways, as shown in figures 1 and 2, or in a manner which is a combi- nation of the two motions :— Fig. 1 shows a purely transverse vibration, while in fig. 2 the motion is solely one of oscillation about the centre of eravity. In the transverse vibration the conditions are clearly the same whether the rotor body is a disk or is cylindrical ; but in the case of the oscillation, the motion, owing to the gyrostatic effects called into play, depends both on the proportions of the rotor and the speed at which it is running. The speed at which transverse vibration becomes a maxi- mum we shall call the “first critical speed,’ and that at which the oscillation becomes a maximum, the “second eritical speed,” as the latter is in all practical cases con- siderably higher than the former. The following is a general outline of the treatment adopted and the conclusions reached :— Section 1 deals with the vibration of a rotor when not running, and a relationship is deduced between the fre- quencies for the transverse motion and for the oscillation which we shall call respectively the “stationary first critical speed,” and the “ stationary second critical speeds.” The second section deals with the transverse vibration, frictional resistance being ignored. It is first shown that there appears to be no foundation for the frequently ex- pressed view that there is a possible region of marked vibration at +, times the first critical speed, as such a conclusion can only be reached through an incorrect assump- tion with regard to the conditions. It is then shown that the motion or vibration is a circular whirl about the statically deflected position of the shaft, and that this motion reaches a maximum at a speed equal to the stationary first critical speed. The magnitude of the whirl is proportional to the amount by which the machine is out of balance, so that the main vibration here dealt with should disappear with good balancing. The action of gravity is then gone into more fully, and it 124 Mr. G. Rodgers on the Vibration is shown that in addition to producing the ordinary static deflexion, the action of gravity is such as to cause a double frequency ripple in the whirl which would tend to reach a maximum at half the first critical speed. The magnitude of this ripple is, however, proportional to the square of the amount by which the rotor is out of balance, and would therefore fail to appear in a well-balanced machine. In any case the effect is very small. It is then shown that a rotor with bi-polar asymmetry, such as exists in a rotor slotted for a two-pole winding, may show a double frequency vibration at half the critical speed even when the rotor is perfectly balanced, so that such a machine might vibrate at half the critical speed even when it would ren perfectly at the full critical speed. Vibration arising from this cause could not, theretore, be rectified by balancing, and this is the only case met with where improved balancing would not effect an improvement in the running. This case is gone into in some detail, and it 1s shown that the motion here also is a circular whirl of double frequency, that is, of twice the speed of rotation of the machine. If, in addition, the machine is out of balance, a triple frequency effect might appear, but is not likely to do so. The effect is then discussed of lack of proportionality in the deflexion of the shaft and again the possibility appears of vibration appearing at half the critical speed, but only if the machine is not properly balanced. The effect is then gone into of fluctuations in the angular velocity through variations in the driving torque, and of resonance between the rotor and the foundations or other masses outside the machine, from which it appears that marked vibration might appear at almost any speed through either of these causes. The effect of friction on the transverse vibration is then discussed, and the results are given for the case where the frictional resistance varies as the first power of the speed, and also where it varies as the second power of the speed, the latter being more probably in accordance with the facts than the former. It is shown that the maximum vibration appears in both cases at a speed equal to the stationary critical speed, also that the phase difference between the force due to the out-of-balance and the displacement depends on the amount of friction, and also on the speed. If the frictional forces vary as the square of the speed, as is probably the case, the angle varies also with the amount by which the rotor is out-of-balance. Some effects of viscosity of oil in the bearings, and of different bearing clearances are then gone into. be i as and Critical Speeds of Rotors. 125 In Section 3, the oscillatory vibration is dealt with, taking into account the gyrostatic effects when the machine is rotating, but ignoring the friction in order to keep the expressions as simple as possible. It is there shown that the gyrostatic effect causes the point of marked vibration to occur ata higher speed than would be the case if the machine were not rotating, and simple rules are given for calculating this vibrating speed. An example is added to illustrate the method of working the rules given. Much of the work on the main transverse vibration and the main oscillatory vibration bas been dealt with in various forms by Chree, Stodola, Morley and others, and the solution for the transverse vibration with friction depending on the first power of the speed has been given by H. H. Jeftcott (Phil. Mag. March 1919), but the ground covered by the remainder of the paper, particularly the question of sub- sidiary critical speeds, does not appear to have received much attention ; there is, however, in ‘ Hngineering’ a dis- cussion where subsidiary critical speeds are touched on, arising out of a paper by W. Kerr in that journal (Feb. 18th, 1916). Srcrion 1.—STarioNARY VIBRATIONS. A. Transverse Vibrations. 1. If M is the mass of the rotor body (the mass of the shaft being being neglected), and we assume the rotor to be perfectly balanced, the shaft will, when not rotating, show a deflexion measured at the centre of gravity of the rotor of Pee eC ee eee 8 CL) where a is the force required to produce unit deflexion. The method of working out the static deflexion of the rotor for ~ actual cases is well understood and the value of o can be got from the deflexion diagram. 2. If now the rotor is set in vibration in a vertical plane, the motion is represented by the following equation (using d? fluxional notation, where # is written for “7 and y for at? dy etc.) de? Mij+éy+My20:- 0: 2. (2) 126. Mr. ©. Rodgers on the Vibration The solution is yatiain(s/Zeom) M2, where N, and y; are constants the values of which depend on the initial conditions. The vibration therefore takes place about the statically deflected position as a centre, and with a frequency of vibration of a, where 2a a=a/ x, Ms spi ol ee This vibration takes place in a vertical plane and may be considered as the resultant of two vectors rotating in oppo- site directions, each with an angular velocity of fe 2 aE M o and M are expressed in c.g s. or f.p.s. units, this angular velocity will be in radians per second and since from (1) U is numerically equal to a the speed of either of these 0 M 60 g vectors in revs. per minute will be ie 9 If, further, g and yo are in c.g.s. units we have the conn ost 2 300 Yo Yo where yp is the static dellenionn in cm. 3. It will be seen afterwards that, as is well known, this formula gives the first critical speed in R.P.M. ; this is to be expected, as the out-of-balance forces will then resonate with the natural free vibrations, with the result that the latter will become of considerable magnitude. Jide t= = 5 = -approximately, . (5) B. Oscillatory Vibrations. 1. If the rotor is twisted about its centre of gravity so that the deflexion is in a vertical plane, and is allowed to oscillate freely, the motion is represented by Babich =O; ene i an where B is the cross moment of inertia, that is, the moment of inertia of the rotor about a line through the centre of gravity at right angles to the shaft, yr is the angle through and Critical Speeds of Rotors. 127 which the axis of the rotor at its centre of gravity is deflected from the stationary position, and « is the torque required to produce unit angular deflexion. The solution is p=Nosin(a/Ke—m) +r Neate ener Ge) where N, and y2 are constants the values of which depend on the initial conditions. The frequency of the oscillation is therefore itiof ie hiongl Onn Bo on where a=al *. ee eee =. (8) As already indicated, we cannot at once deduce from this what will be the actual second critical speed, owing to the gyrostatic effects, but the result is of importance, as it simplifies the calculation of the actual second critical speed, as will be shown later. We shall in what follows call ¢, the stationary second critical speed. 2. It should be pointed out that there is a simple relation between c, and c, which greatly facilitates the calculation of the stationary second critical speed in those cases where the centre of gravity of the rotor is midway between the bearings. If 2] is the distance between the bearing centres and P, the force exerted by the deflected shaft on either bearing, np = ZBL The angle is very small so that the force P, is the same as would be required to depress the shaft through a distance al if the rotor were held rigidly. Now we have seen that the force Mg at the centre of gravity causes a transverse deflexion of Yo=r— eS and as is small, yo=vl, so that Mg=ocowl, also kyp=2P,l and P,=3Mg; therefore ky = Mol, so that =o vier), nt and (8) becomes mia. as SO aren pee a) 123 | Mr. C. Rodgers on the Vibration and at B=Mzk,’, a am = hp M ’ and comparing with (4) we thus get: (a a 9 oe gf 12) We thus find that | First critical speed (transverse vibration) Stationary Second critical speed (oscillation) ” __ Radius of Gyration for the cross moment of inertia Half the distance between the bearing centres This is a useful formula for calculating the stationary second critical speed when the first is known, for cases where the centre of gravity is midway between the bearings. It shows that with cylindrical rotors of this type the second critical speed must always be considerably above the first, and the only instance in normal designs in which the second critical speed could be lower than the first would be that of a flywheel mounted on a short shaft. Secrion I1.—TRANSVERSE VIBRATIONS—FIRstT CrITICAL SPEED. ~ A. Neglecting Frictional Resistance. 1. It will simplify the treatment of this question if we first consider the case of a rotor unimpeded by frictional resistances set up by the air and then treat separately the effects produced by friction. The conditions obtaining when a rotor is not perfectly balanced and is rotating are illustrated in fig. 3, where O represents the position of the centre line «f the bearings, and C the deflected position of the centre line of the shaft, while G shows the position of the centre of gravity of the rotor. O thus gives the undeflected position of the shaft centre line and OC=r the shaft deflexion at any instant, while CG=e is the error in the centering of the rotor ; Mg is the weight of the rotor acting vertically downwards. The rotation of the rotor about its centre line, 2. e. the rotation imparted by the prime mover, is represented by. the motion of G around G, ¢. e. by the rate of change of 0. and Critical Speeds of Rotors. 129 The whirling of the rotor is represented by the motion of C about the undisturbed position of the shaft centre line, 2. e. by the rate of change of a. The “ vibration” of the rotor is judged in a general way by the vibration of the bearings as felt when the hand is applied to them. The force on the bearings is that applied along OC by the deflexion r of the shaft, and vibration of Fig. 3. the bearings arises through the varying position and magni- tude of OC; these in turn are due to the motion of the ceritre of gravity G. 2. If the machine is steadily rotating it might at first sight be thought that OC and CG would be in the same straight line, so that G would be steadily revolving together with C about the undisturbed position O of the shaft centre line with an angular velocity n say. At the same time the deflexion OC=r might be changing its value and (neglecting the weight of the rotor) the motion would thus be given by Mr—Mn?(r+e)+or=0, or putting o/M=c,’, P+ (ce —n?)r=n’e, the solution of which is : Te eee ne r= N,sin (./ = Seer eRe? : cy? —n? 11 cy —n*’ / where N, and y; are constants. We should thus conclude that » would become unlimited Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. K 130 Mr. C. Rodgers on the Vibration in magnitade when the angular velocity n is equal to ¢ or 4), ap and, further, that the variation in the magnitude of 7 consists of a free vibration having a periodicity of 1 V4 A Ser, N ef —n?’ This result for the periodicity of the free vibration would lead to the conclusions that when the machine is not rotating (n=0) the periodicity is c, the same as for the stationary transverse vibration, and that when running at the critical speed (n=c,) the periodicity is zero. There would thus be some intermediate speed where the periodicity of the free vibration corresponds with the running speed, and resonance might take place. This would occur when ¢2— nan? n “1 or SS V2 We should thus be led to expect marked vibration when the running speed is 45 x the critical speed. This conclusion and the argument on which it is based are, however, erroneous. In the first place, the assumption is made that w or @ is constant and further the condition is omitted that, as all the forces pass through O, the angular velocity about O must be constant, or °>9=h, say. The correct equations for the free vibrations are thus : Ga eR Ie 0, PA=h. | This does not admit of direct solution *, and it is simpler to use rectangular co-ordinates, as we shall now proceed to do. * The solution is, however, well known and is given in books on Dynamics dealing with Central Forces :—- If p is the length of the perpendicular from the centre of force on the tangent to the path, it is known that h? dp PON eae ps dr ‘ ae h? ; giving — =a constant— or", pi which is the pedal equation of a central ellipse. and Critical Speeds of Rotors. 131 3. With the notation given in fig. 3, it will be seen that the position of the rotor is completely defined by the co- ordinates x, y, and @ (wand y being the co-ordinates of G) and only three equations are required to express the motion fully ; the value of the whirling angle a follows from the magnitude of the other co-ordinates. The force exerted by the deflected shaft is or, the com- ponents of which are —o(v—ecos@) along OX and —o(y—esin@) along OY. Resolving along OX and OY and taking moments about G, we thus have: Meso (ce (Cosiy JON yh ieee, (a8 7 (13) Mi+o(y—esin@)+Mg=0, . . . . (14) Mi.20+ce(asin@—ycosO)=0, . . . (15) where k, is the radius of gyration of the rotor about the longitudinal axis through the centre of gravity. In practice the rotor is driven at an average angular velocity » say, which will vary from constancy only by small amounts which we shall find later are negligible. Assuming as a first approximation that the angular velocity is constant=@, so that @=qt, (13) and (14) then become, writing ¢/M=c,’ as before, HO eye COBIG ais a) He Me CLG) Uti yy OSU WG.) tiie duels) GL) while (15) becomes an identity. The solutions are € red C)7e w=N, sin (cé—y1)+——5coswt, . . . (18) Cea) COS (Cyt — 1) + ae wt —g/cy’. . (23) 1 — 5. These equations are the same as for a perfectly balanced rotor with a weight attached to it of such small value as not to affect the position of the centre of gravity, or the down- ward pull due to gravity, but producing a force of Mo’e. In other words, we can treat the unbalanced rotor as if it were a perfectly balanced rotor with a force Mw’e attached to it, and as this mode of presentation is easier to follow than the former, we shall employ it in the remainder of the paper. 6. It will be seen that the solutions for the motions of the centre of gravity and the centre line of the shaft are the C,7e same, except that the former has an amplitude —, — ,, and Cm @ the latter an amplitude of = so that they differ by the C7 == } amount e; this shows that OC and CG are in the same straight line when not running at the critical speed. 7. The solutions show firstly that the motion takes place about the position Mig ig 99 (oR Cy™ which we have seen is the statically deflected position of the centre of gravity of a perfectly balanced machine. It is sometimes contended that as the speed increases the rotor shaft tends to straighten out, but there is no indication in the present treatment that this is the case. &. The free vibration “«=N,sin (¢,¢-y;), y= Ny. cos (q¢— 1), is the same as for a perfectly balanced rotor and has a and Critical Speeds of Rotors. 133 frequency the same as the frequency for stationary vibra- tions, and, as we shall presently see, if expressed in R.P.M., is the same as the first critical speed. It is, therefore, inde- pendent of the speed of rotation of the rotor, and there is no possibility of resonance occurring at V9 x the critical speed as suggested by the erroneous method mentioned earlier in this section. ‘The free vibration itself is thus represented by two components having the same frequency, but different amplitudes; it is therefore a central ellipse, the centre Mg Sar It will be shown later that the free vibration is damped out by friction, so that it has no importance in practice. 9. The forced vibration for the centre C of the shaft is given by being at the point y= — we i c2—o Cos at, ‘ e 5 = . (24) Hy Dre Os i. On = ae sin ot — 9] cy 5 A : c (2: ) This naturally has the frequency corresponding to the angular velocity of rotation w, and has a maximum value of 9 @~e . for each axis. 6 2 The motion of the centre is thus a whirl, the radius or amplitude of which is proportional to the out-of-balance and is zero when the rotor is perfectly balanced. A perfectly balanced rotor, therefore, cannot whirl in the manner ex- pressed by equations (24) and (25). 1 The amplitude of the whirl is also proportional to —;—, e - ! : Cy —-@® and it thus becomes a maximum when oO = Se Oya ([he sign + merely indicates that the rotation may be in either direction.) This value of w gives the first critical speed, which is thus the same as the stationary frequency for transverse vibra- tions. Reasons will be given later why the radius of whirl does not become infinite at the critical speed, ¢.¢., why the shaft does not break when the rotor reaches this speed. ° 134 Mr. C. Rodgers on the Vibration 10. At the critical speed where w?=c,? equations (20) and (21) become : di + ci?" = ¢,"e cos yt, YtoPy=cesin oyt—g, which admit of a solution not involving infinite values, namely, (oe v=t— sin Cyt 2 C1é F TT = cos( ot - Ae os C1eé cos yt —g/cey Cc . =1Fsin( et 5) gore ere Equaticns (26) and (27) thus give the motion at the first critical speed when friction is ignored. They show that the component along each axis has an amplitude which con- tinually increases in proportion to the duration of the motion, in other words the motion at the critical speed is a spiral of continually increasing radius. 11. From (24) and (25) it will be seen that the phase difference of the motion with respect to wt changes from zero to 180° as w passes through the value ¢, 7. ¢., as the speed passes through the critical ; (26) and (27) show that at that speed the motion lags behind wt by 90°. It will be seen later that when friction is taken into account, the lag increases gradually as the speed increases, being still 90° when o=(}. | 12. Up to this point we have treated the angular velocity @ as a constant=q@, as on the average it will he in practice. Suppose now that it varies slightly from constancy, so that the angular position wt becomes wt+u, where u is so small that its square can be neglected, and we may write sinu=w and cosu=Il. We then have cos@=cos@t—usinwé and sin 0=sin wt +u cos at, also O=ii. Substituting these values in equations (20), (21), and (15), and writing i =O ee before, we get é+o’e=ew7%ecos@t—o*eusinwt, . . . (28) YtocP=yow'e sin ot+@*cucoswt—g, . . (29) Mit oe(w sin wt —y cos wt + xu cos at +yusin@t)=0. (30) and Critical Speeds of Rotors. 135 As a first approximation we substitute in (30) the values already found for # and y, as given in (24) and (25), for the forced vibration and thus obtain (neglecting wu in comparison with 1 radian) : 2 ee @ 1 — giving for the forced vibration (the free vibration may be ignored as it will be damped out) : . e(¢,?— 0”) ae wee)? — wk (cr— w”) a9 COM@in Ne « - (32) At the critical speed w=c,, and u=0, 2.¢., the angular velocity of rotation is constant ; at other speeds than those in the neighbourhood of the critical, the term we?c,? may be neglected, as e? is much smaller than /,?, so that eb) ae ee é Boron wt. . . a a (33) The variation in the angular velocity has thus the same frequency as that of the rotation itself, but is very small in magnitude, as will be seen from the following figures. For a turbo-generator rotor balanced to about 1 oz. at radius ky per ton weight of rotor, e/k, will be about 3x 107°, and for a machine to run at 3000 R.P.M., &, will be about 90 cm., and @ is, say, 277 x 50, U= 3x10 mia teks eos at ~ 50. ~*~ 4m? x 50x 50 =6 x 10-* cos at, that is the vibration is very small, the amplitude being only 6X 10~ of one radian. It should be noted that w is sao genoa to eand to q; this variation therefore arises through the action of gravity on the rotor when not perfectly balanced, and the variation will be absent if the balance is perfect. Substituting the -value of wu in (28) and (29) in order to find the effect of the irregularity on the displacement :— al ui+ec?r=w’ecos wot + 7.29 sin wt cos wt, | 2 Ne e ¥ + ¢7y=o7e sin wt — Ea 9 cos? wt —g, ‘ Uy 136 Mr. C. Rodgers on the Vibration or d+ ¢x=w'e coswi+ <— g sin 2at, € 2 ky ee ; 1 1 @2 Yto-y= we SID w Seg 082 Zot — 9 Wo ; The solution is (for the forced vibration) w’ecoswt 1 é i 4 ; a= moog ar 9 hee hey? SIN 2@t, 9 S.. e aeoe) wesinat 1 é 1 a / 5 = = Ses os 2 Jae il 35 Pow? 2 be eee +s 2 fe =). (35) The irregularity has thus two effects on the main whirl : firstly, the static deflexion is increased by a small amount, and secondly, there is superimposed on the main whirl a ripple of double frequency, which rises to a maximum at half the critical speed. But the effect is very small, and may not be noticeable ; in any case, as the double frequency effect depends on e it cannot appear when the machine is well balanced. . There is, however, the possibility that a rotor which is not perfectly balanced may show vibration at half the critical speed due to the action of gravity, although gravity would produce no such effect at the full critical speed. A vertical _ spindle rotor is not of course subject to the action of gravity in this sense, and if it vibrated specially at half the eritical speed, the cause must be sought elsewhere. 13. We shall now consider some further possible causes of subsidiary critical speeds or speeds where marked vibration may appear other than the normal calculated critical speed. 14. An important case is that of a rotor slotted for a 2-pole winding or with a shaft in which a key-way is cut, where the rigidity of the rotor is greater in one direction sham in a direction 90° away, so that if the shaft is rotating, the stiff- ness in the direction of any one of the axes is not a constant o but o +¢€ cos 2wt, where ¢ is small in comparison withe. We shall assume that the rotor is perfectly balanced (e=0) and to simplify the examination shall first consider the vertical motion only. The equation is: My+oy=—Mg— poe 2ot. The first approximation is : and Critical Speeds of Rotors. 137 Inserting this on the right-hand side of the equation and solving, we get Mg ; € a = ———— - 2 t y o 1 mea Wipe ey he jvc/Dls yn) 1 =—g/c {1+ oa gp 008 Bat BP es ei {ee Os) There is thus a double frequency vibration about the statically flexed position, which has a maximum when @ =), that is, at half the critical speed. It is evident that such a motion must have a tendency to arise if a rotor is unsymmetrical as regards its rigidity, for in such a case when the shaft rotates the deflexion will be a maximum or minimum twice every revolution, and if the frequency of the consequent up and down motion is equal to the critical speed there will be resonance ; this will be the case whether the rotor is perfectly balanced or not. Tt is thus possible for a perfectly balanced rotor which would be quite steady at the critical speed to show marked vibration at half that speed. If the normal running speed is above the critical the forces called into play at half the critical speed will be very small and may give no appreciable effect, but if the running speed is in the neighbourhood of half the critical speed vibration might arise. Hig. 4. 15. It is worth while to examine the motion a little more fully as there will evidently be some vibration in the hori- zontal plane also. Let C (fig. 4) be the position of the centre line of the shaft and OA, OB two axes at right angles rotating about O with the same angular velocity w as rotor. Let the co-ordinates of © be a and 6 with respect to OA and OB and wu and v the corresponding velocities along those axes. ee 138 Mr. C. Rodgers on the Vibration Then u=a—bo, v=btao, and the accelerations are along OA: %t—va=a— 2he —aw’, along OB: v+ue =b + 2a0—be?. If the force required to produce unit deflexion in the shaft is o+e along OA and o—e along OB, and we Tee: along OA and OB, the equations are: M(a — 2bw —aw?) + (o+¢)a= —Mgsin ot, M(b + 2a@ —bw*) + (c—e)b = — Mg cos ot ; that is, {M(D?—o?)+ (¢+¢€)}a—2MeDh=— Mg sin ot, {M(D?—o?) + («—e)} 6+ 2MoDa=— Mg cos ot, giving [ {M(D?—o?) +0}?—e? +4M?w?D?] a | = — Mo(o —e— 4”) sin of, [{ M(D?—@?) +0}? —e? +4M?w?D? | | =— Mg(o+e—4o’) cos at, The solution is (neglecting e? in comparison with o) c= = thie) sin at, b= - =o (aee ee ,) eos wt. This gives the position with respect to the rotating axes ; the position with respect to the fixed axes is v=acos wt—b sin at, y=b cos wf+asin of, that is: | tect VEG € ah pe Jaa | SIM De ese (37) iis. «ela € yao fis =e? 008 200 | ial SNR, pee This result is the same for y as obtained in (34) by the and Critical Speeds 07 Rotors. 139 method of approximations ; it shows that there is a similar motion of equal magnitude and 90° out of phase along the horizontal axis, so that the motion is a circular whirl of double frequency, which rises to a maximum at half the critical speed. 16. When the rotor is out of balance the equation for the vertical motion is Mi + oy=Mo’e sin at —Mg—ye cos 2at. The first approximation is given by (25) and inserting this on the right-hand side of the above equation we get: Mij+oy=Mo’e sin ot —Mg+ = € COS 2wt e Moe + 3 SMe? Lin et sin Bat}. (39) The first three terms on the right correspond to the main whirl and the double frequency whirl already dealt with. The last term on the right will give in the solution a triple frequency vibration, viz. : 1 Moe € as i Me eeoMae oe which has a maximum value at 4 the critical speed. This vibration is, however, proportional to e, the out-of-balance force, and cannot arise in a perfectly balanced machine. The remarks made as to the limited conditions under which the double frequency vibration might arise apply with even greater force to the triple frequency vibration as the damping effect of friction will be correspondingly greater. 17. Another case of interest is that in which covers or sleeves are mounted on the rotor, or the rotor has slots in the periphery for an exciting winding, closed by pressed- in keys; the closeness of these force fits will vary with the deflexion, and the deflexion of the shaft may therefore be not quite proportional to the force applied, 2. ¢., the force to produce a deflexion x will not be ox, but say o(a+ea2*), where ¢ is small in comparison with unity. (The expression for the force must contain odd powers of a only as the rigidity is symmetrical, that is, the same numerically for the same numerical value of « whether « is positive or nega- tive; if even powers were included this could not be the case as an even power of .v is always positive even if w itself is negative.) 140 Mr. C. Rodgers on the Vibration The equations then are, putting the small quantities on the right-hand side: a+ ¢,?a4 = we cos wt — ¢,7exr’, YT cy = we sin wt —g—¢,7ey?. Neglecting the small quantities, the forced vibration is as betore given by ®) we | ff == ¢;°— @ COS ot, ° ° ° e ° . (40) ORE | — aa sin wt —9/aq2..° 2 ae) Inserting these values on the right-hand side of the original equation, we get after some reduction :— a + 7a = {we — 3c?ep?} cos wt — 1c,2ep* cos 3at, 2 2 Daye eee Dee 9 + ery = —9 + 61° €yo(p? + yor) + {we + cep (2yo" + Zp”) f sin wt —c¢yep7y) cos 2wt + +¢,7ep? sin 3, c,7e where ~ Pome ; ea) and You ger Solving these equations we get: 2 ee) p L=p | 1-40 OF aH cos wt —1¢/? COs dwt, eae C7 -- Q@? Y= — yo{ 1—eyo(p? + Yo") } a = Ne ce —@" (2y9? + 3p”) ; sin at 2 GCOS Wt cy — 407 pPYo Soneane pemrhee p> sin det. Examining these terms in turn we find that the centre of motion is now at the point e= 0, y= —Yyotl—eyo(p +4?) 5, instead of the point e=0, y=— yo. This indicates that the centre of motion rises, 7. e., the shaft straightens out slightly, as the vibration increases. and Critical Speeds of Rotors. 141 The main vibration, represented by coswt and sin af, shows a slight change in amplitude, but as before the maximum oecurs at the critical speed. The term in cos2t indicates there is a double fre- quency ripple in the vertical motion y (but not in the , 26 ° . . C7 horizontal component S) having an amplitude aa a 5 P'Yos ae > C) € wo € 1 a) Yo This rises to a maximum G2 — that is, a des at half the first critical speed (when the amplitude changes sion) and again at the first critical speed (when the ampli- tude does not change sign). Noticeable vibration may thus occur at half the critical speed, but it will take” place principally in the vertical plane. Both components show a triple frequency vibration ex- pressed by cos3@é and sin 3et, which reaches a maximum at one-third the critical speed, and the amplitude of the vibration changes sign at that point. This vibration also has a maximum value at the critical speed. Points of marked vibration due to lack of proportionality in the deflexion can thus only show themselves when the machine is out-of-balance, and if they become appreciable at all will only occur at half or one-third, ete., of the critical speed. If, however, these fractions of the critical speed correspond to low running speeds, the forces may be so small as not to produce any noticeable effect. 18. It thus appears that subsidiary critical speeds are only to be expected at half or possibly one-third of the calculated first critical speed, and only then when the subsidiary critical ‘speed is high enough to make the forces appreciable—for example, in the case of a turbo alternator when the speed indicated by the calculation approaches the running speed. All these effects should disappear with perfect balancing, excepting that due to lack of uniformity in the resistance of the shaft or rotor to bending in directions perpendicular to its axis, such as might arise “through two-pole slotting of the rotor or through a key-way in the shaft. The forces tending to produce vibration are small, and the vibrations arise through a kind of resonance ; as there is a good deal of damping due to air friction and to the move- ment of the shaft in the bearing where the oil exercises a strong damping action, the vibrations may not arise at all. This question is gone into more fully in a later section. 19. We have now to consider some cases where resonance may arise from causes outside the machine itself, and two 142 Mr. . Rodgers on the Vibration classes may be noted, firstly where there is an irregularity in the torque applied to the shatt, and secondly where there is resonance with masses outside the machine. 20. Irregularities in the torque driving the machine may arise, for example, through variation in the steam admission or through a fluctuating electrical load. The result of fluctuation in the torque will be a corre- sponding fluctuation in the angular velocity of rotation so that the angular position, instead of being o?, will be wt+esinpt, where € is a small angle and p is an angular velocity corresponding to the frequency of the disturbance. Then cos (esin pt) =1 and sin (esin pt} =esin pt. The equa- tions then become, taking the small quantities on to the right-hand side : CONG 9 9 " ° .s % + ¢°u =e (cos wt —esin pt sin at), jto?y=o’e (sin wt +e sin pt cos at)—g; that is, u+tcPa=ore{coswt+e/2(coso+p.t—cosw—p.t)},. (42) . (48) The main vibration is the same as before, but there are two small vibrations superimposed ; the one has a frequency corresponding to a +p and a maximum when w=c,—»p, the other a frequency corresponding to w—p and a maximum when @=¢,+ 7p. This shows that the vibration may have a maximum at speeds corresponding to the sum of and to the difference between the critical speed and the speed corresponding to the frequency of the disturbing fluctuation. So that if dis- turbed running show itself at such a speed that it cannot be otherwise explained, a cause may be sought for in this )y yj + cyy = @e{sin @t + e/2(sin op (esos o- p ; t) i _direction. 21. The other variety of resonance mentioned is that where, for example, the foundations are not sufficiently rigid and the machine as a whole is vibrating so that there is resonance between the rotor on its shaft and the machine on its foundations. A similar case would be that of a machine rotating in or near a building which itself shows marked vibration, possibly in certain parts only, corresponding to the vibration of the machine. Both these cases are similar in principle and may be illustrated by supposing the whole machine to be mounted on foundations having some elas- ticity. If then M, is the mass of the rotor and M, the and Critical Speeds of Rotors. 143 effective mass of the machine and that part of the founda- tions which moves with it, and the forces required to give unity deflexion are in the two cases o, and ay respectively, the equations of motion are as follows : My, + om (yy = Y2) 2s M,o7e sin pt, Moijo + O2Y2—01(Y1 — Y2) = 0. giving for the amplitude of the forced vibration we (¢1?m + 9? — w”) y= es a Wied” CAA Tee wo? (Cy? + Co? + C47) + C475” — ‘ O71 9 Oo M, mhere ¢¢=———, ¢-=—,.) m= — M,’ M,’ M, Points of marked vibration may thus occur at either of two frequencies given by putting the denominator = 0 ; these frequencies will therefore depend on the ratio 2 as 3 2 well as on c, and c., and may thus have almost any values. For example, if cp=c,; and m=0°2,12.e., the mass of the machine and foundations is five times the mass of the rotor— ype ed Be Se) w* — WoC? X 2°2 + ¢;*’ and a maximum occurs when w=c, X12) or 6, x 0°80, that is, at speeds 25 per cent. above and 20 per cent. below the calculated critical speed. If M,, the mass of the machine and foundations, is very large in comparison with Mj, the mass of the rotor, the denominator is very nearly equal to (w—c,) X (o—c.), which shows that in such a case the two speeds where marked vibration may occur nearly correspond to the natural fre- quency of the rotor and of the machine and foundations respectively. But as the numerator is also small the vibra- tion might not appear if considerable friction is present. If vibration should occur when w=c,//2, which is, as mentioned above, sometimes thought to be a critical speed, this might indicate that there was resonance with the founda- tions or some structure outside the machine, in which case, 144 Mr. C. Rodgers on the Vibration putting w?==¢,7/2, 67 —2 {¢,7(L+m) +7} +4¢2=0, that is CA, ae Vi 2D so that if, for example, M, were large in comparison with M, and m is therefore small, Pao A= 6, i and ds ; : Me (G; yo 7,7 that is, Mus‘ and Mys"~’y, where mw is a constant. 3. In the particular case we are about to consider, n=1, and the components are therefore Mud and Mpg. and Critical Speeds of Rotors. 145 The equations of motion are therefore Mé+Mypet+or= Moe cos at, My + Mpy+oy=Mo’e sin wt —- My. Electrical engineers will notice the similarity between these equations and Lg+Rg+ cq=E sin wt + Ko, I: which holds for a circuit comprising an inductance L, a resistance R, a capacity K, an alternating E.M.F. of maximum value H, and a steady H.M.F. Hy, ¢ being the charge in the condenser at any time. Thus, mass is equiva- lent to inductance, capacity to deflexion per unit force, and applied H.M.F. to applied mechanical force. The solutions of these equations are, as is well-known :— g= Ne *" sin (pt— $) + z par gets hs ) Ft /Kon tary ( 1/Ko—Lo, EN ah mn emEnS Ay ca ic tnt ky eng Weim ho ey Cede) y=Ne ‘sin (pi—¢@) Mo’e _, Meo sin (ot —tan Fane), = WV { (o— Mo’)? + M?u?w?} —Mglc, (46) that is, SS y=Ne #T sin pt—o we : % @ Ti Veo)? 4 we? sin (wt —tan : aS — gfe’, 47 where ey OL, i 2M (47) art: Mp’ p=v 1/LK—1/T? or Vo/M—1/T?, Cy = stk or VW o/M. There is therefore in both cases a free vibration having a frequency slightly less than the natural frequency of the system, but independent of the frequency of the applied W.M.F. or of the speed of the rotor. This vibration is damped out by friction. Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922, L 146 Mr. C. Rodgers on the Vibration The forced vibration is permanent and has a value which is proportional to the applied H.M.F. or out-of-balance force. The vibration is a maximum when o=«, that is the critical speed is the same as the stationary critical, as in the case where friction is ignored; the amplitude of the vibration at the critical speed is : or a that is, is equal to the applied E.M.F. or out-of-balance force and inversely proportional to the resistance. The frequency of the forced vibration is the same as the frequency of the E.M.F. or of the rotation, and the charge or displacement lags behind the E.M.i’. by an amount depending on the frequency or speed and on the capacity and inductance or elasticity of the shaft and mass of the rotor. The lag is zero when the frequency is low, but increases to 90° at the critical speed, which, as will be seen, is that corresponding to the natural frequency of the system, while at very high speeds the lag increases to 180°, in other words the force is in opposition to the displacement. The change is similar to that occurring when there is no friction except that in the present case the change is gradual instead of taking place suddenly at the critical speed. It will be noted that in both cases the vibration takes place about the statically deflected position as a centre. It is evident in both cases that a large static deflexion would increase the tendency to break down, in the one case by puncture or flashing over of the condenser, and in the other by fracture of the shaft. 4, If the resistance to whirling is proportional to the square of the speed, that is n=2, the equations-are : + pst + c2u=w'e cos wt, i + psy +¢°y=o'e sin wt — 9. The free vibration (7. e. the vibration when e=O or the rotor is perfectly balanced) cannot be expressed in simple terms, but as it will be damped out, as before, it is not of interest. The forced vibration (2. e. the vibration due to the rotor being out-of-balance) is given by e—Rieos(@t—O).4 o) os) - deem y=R sin (@t—d)—g/c",. - - . (49) and Critical Speeds of Rotors. 147 where R is the radius of whirl of the value : ae | pan eee 50 Vi V (cr — 07)! + 4e?otpw' + (¢7—w?)”} aM, and tan ¢6= Le Pe 2S (51) The lag of the displacement behind the force is in this case proportional to the actual deflexion, and in this respect differs from the result obtained in (47). This is of interest as it shows that since the radius of whirling is for a given speed dependent on the out-of-balance, the phase lag will be smaller the more perfectly the machine is balanced ; in the former case where the friction varied as the first power of the speed, the lag was independent of the amount of out-otf-balance. The maximum deflexion occurs, as befcre, when a= ¢,, that is, when the speed is equal to the stationary critical speed. 5. It is impossible to draw any conclusion from these formule as to the real angular advance corresponding to a given speed, as it is not known how the frictional resistance varies with the speed. We can, however, say that if the machine is rotated first in one direction and then in the other, the position corresponding to the out-of-balance will be mid-way between the points of maximum deflexion. When balancing a machine in the running condition it is usual to hold a pencil or chalk against the shaft so that a mark is made on the shaft at a point corresponding to the maximum deflexion. If there were no friction and the speed were not the critical speed, this mark would be in phase with the heavy side of the rotor below the critical speed, and 180° out of phase with it if above the critical speed. But it will be seen from (51) above that the actual position of the mark depends both on the amount of friction and on the amount of out-of-balance. At the critical speed the heavy side of the rotor should be 90° out of phase with the mark on the shaft, but the actual position will be uncertain, as the angle varies rapidly with departure from the critical speed, and it is not usually possible to judge exactly when the machine is running at the critical speed. 6. There is another reason why the position of the mark on the shaft is somewhat uncertain. Referring to fig. 3, if we ignore all other vibrations than that corresponding to the variation in ¢, we get by taking moments about G: Mked+oresind=0. . . . . (52) L2 148 Mr. C. Rodgers on the Vibration If ¢ is a small angle this becomes : Mh 2g +orep=0, the solution of which indicates a periodic motion having a time of vibration of Mk? hore ky = 2or oe bene (53) In an actual machine for 3000 R.P.M. we shall have figures, of the order of:-%=90 cm, Lje=— ae ¢;= 27 x 30, say, while » may be of the order of 1 mm., so that P= 27; ah soar 50x3x 104 ~ Qer x 30 O-1 == 1-410? see. or about 2 mins: As the time of vibration is very long in comparison with that of the other vibrations occurring, it will be almost. unaffected by the latter, and the assumption that the other vibrations can be ignored, which was made in deducing (52), is therefore justified. For lar ger values of ¢ corresponding to less perfect balance and for deflexions of greater magnitude, T will be corre- spondingly less, and will be oreater the more perfect the balance. If friction is ignored, r becomes infinitely great at the critical speed and I’ becomes zero, and although this can never be the case in practice, it is lore that T’ may have a value of two mins. or more down to something considerably smaller. In other words, if the rotor is disturbed from its position of equilibrium by any chance external cause, it may take a considerable time to settle down, and during that period the position of the mark on the shaft. will vary considerably from its normal position. 7. At the critical speed the lag is 90°, and the vibration is also a maximum, but the sharpness of this maximum will, as indicated above, depend on the frictional resistance to whirling. In addition to this it will also be influenced by the condition and design of the bearings, as the oil in the bearings exercises a considerable damping influence and also introduces a further complication as follows : When the speed is low and the vibration therefore small and Critical Speeds 07 Rotors. 149 in magnitude, the film of oil in the bearings will allow the shaft a certain amount of play ; this will increase the effec- tive length of the shaft and ion er the critical speed; with increasing speed the vibration will therefore start up fairly smartly. As, however, the speed increases and the vibration becomes greater, the shaft may bed hard up against the bearing bush, and increased deflexion will decrease the effec- tive length of the shaft, and so raise the critical speed. As the speed is further increased a similar state of things is gone through, so that at a certain point the vibration will die down more quickly than if there had been no film of oil. The effect of the oil in the bearings is thus to give an added amount of friction to whirling, and at the same time flatten the maximum peak of the vibration, that is, the vibration will start up and cease fairly smartly and remain more or less constant throughout a fair range of speed. If, however, the film of oil is sufficiently thick or the balance sufficiently good, the vibration may not show itself at all, although it might do so with the same out-of-balance, if the film of oil were thinner. Section LI1.—OsciLLAToRY VIBRATIONS—SECOND CRITICAL SPEED. 1. Oscillatory vibration may arise in two ways, either through lack of balance or through vibration transferred from the transverse motion. 2. The lack of balance referred to is of the skew type, that is, is equivalent to a pair of weights at opposite ends of the machine, and 180° apart, giving an out-of-balance couple when the machine rotates; such an out-of-balance will not show itself when the machine is being statically balanced on knife edges, and can only be corrected through observations when the machine is running. 3. Vibration can be transferred from the transverse motion only when the machine is unsymmetrical in the sense that a force applied to the centre of gravity at right angles to the shaft gives a displacement which is not parallel to the centre line, that is, in those cases where, on the static deflexion diagram, the shaft in the deflected position is not parallel to the centre line of the bearings. 4. The form of out-of-balance mentioned will produce a couple rotating with the machine, that is, a couple alter- nating with the frequency corresponding to the running speed. Vibration transferred from the transverse motion may, however, be of the frequency corresponding to the 150 Mr. C. Rodgers on the Vibration speed of the machine, but may also be of double frequency arising through any of the causes we have discussed. Further, when the normal frequency oscillation has estab- lished itself, a double frequency oscillation may start up owing to bipolar asymmetry or some of the other causes mentioned in connexion with the transverse vibration. It is therefore necessary to consider in the oscillatory motion forces both of the actual frequency of rotation and of double frequency. 5. We have found that the transverse motion can, with sufficient accuracy, be considered the same as for a perfectly balanced machine with an ‘out-of-balance force attached to it. In the same way we shall treat the oscillatory motion as being due to an out-of-balance couple of the frequency corre- sponding to that of rotation or a multiple of that frequency acting on an otherwise perfectly balanced machine. 6. In the diagram fig. 5 let G be the centre of gravity of tne rotor and GL the direction of the centre line of the rotor Fig. 5. | ¥ twisted from its normal position by an angle w= LGZ, where | GZ is the direction of the centre line when not vibrating. The direction cosines of the centre line GL with the axes GX, GY, and GZ are respectively £, 7, and ¢; if L is a point at unit distance along the shaft from the centre G, then &, 9, and € are also the co-ordinates of the point L, as shown on the diagram. If the moments of inertia of the rotor about the shaft centre line and about the line at right angles to it, through and Critical Speeds of Rotors. LoL the centre of gray ity, are A and B respectively, the angular momenta are * about GX, hy=B(nf—f) + Aw€, about GY, h,=B(cE—£¢)+ Aon, about GZ, h;=B(£)—7né) + Aol. In the actual case the angle y is very small, so that we can put siny=w, cosy=1, and ¢=1, €=0, also the preducts £7 and n£ are both negligibly small. We thus get : k= —Bn + Ao€, ho=BE+ Aon, h;= Ao. * Another and perhaps more legitimate way of deducing these equations is as follows :— If 0, ws, and 3 are the instantaneous angular velocities about moving axes +X’, GY’, and GL fixed in the rotor and moving with it: consider the stant when GX’ is perpendicular to GL and GZ (ef. fig. 5), and let 9 be the angle between the planes LGZ and YGZ, Then o,= =H. w,=8 sinw, and w,=w. The angular momenta about GX’, GY', and GZ are Bw,, Bw», and Aw, The angular momenta about the fixed axes GX, GY, and GZ ae: 2i=Bi w,cos 0+, cos W sin 6++ Aw, sin sin 8, h,=B{ —o,.sin 9+w, cos cos 0} + Aw; sin W cos 8, hs = Bi — we sind}+Aw,cosw; that 1 h\= B{—v cos 9-+6 sin Y cosW sin 8} + Aw; sin ~p sin 8, ho= Bf wbsine+ésin WJ cosw cos 9 + Aw, sin Y cos 4, h,=—B6 sin? J+ Aw, cos v. Aljo sinwsin@ and c= W cos W sin 0+0 sin W cos 50, n= BUS ON? and y= By conn eon Oi: @ sin Wsin @, g=cos y. and g=—wdsinw; so thit no—ln=—w cos @+6sin cosysin@. gé—tZ= p sin 0-+-8 sin w cos w cos 9, | in—ni= —@ sin? vy. Fy substituting these values in the equations for Ii, h,, and h,, we obain the relations given above. ———————eEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeeEeeEeEe ——— 152 Mr. C. Rodgers on the Vibration The force of restitution of the shaft is : about GX=xn, ott GY=— «6, about GZ =0. If the out-of-balance couple is tw’, the moment of the couple is: about GX= —7w’ sin pot, about GY =7o? cos pat, about GZ =nil, where p= =1 or 2. The equations of motion are thus : —Bij—«n+ Awé=—rtw’sin pot, . . (54) BE+KE+Aon=Twcos pot, . . . (55) Ae Ch) ae or putting D for «/dt, (D? + ¢?)n —moDE=Tw?/B. sin pot, (D? + 62) £ + mw Dn = re0?/B . cos Ue Aw=0, K A where ¢,?= B and m= B: - The last equation A®=0 gives .«=const., that is, there i is no Huctuation in the angular velocity of rotation. The other equations give D? +0)? + m?w?D?! E=7o?/B §{ (co? — pw”) —mpw? coi pot, s ie PO 5 COvP and a similar equation for ¢ in terms of sin pat. 7. The free vibration is not of importance, as it wil be damped out as before, but it is of interest to exami its value as illustrating the effect of gyrostatic action o1 the motion. The free vibration is of the form “x=N,sin (gt—¢;); y=N.cos (gi—¢do), where Nj, dy, No, o2 are constants and =—lLime-+ J mo b4e7to oe 2 167) and Critical Speeds of Rotors. 153 There are thus two natural frequencies of whirling, depending on the direction in which whirling takes place. For example, if the machine is running at what we shall ; fea presently see is the critical speed, namely, mo=co= ve : there are two possible frequencies of whirling for the free vibration, viz.: g=1°618 C., or 0°618 C,. 8. The forced vibration is given by o ta" /B ‘ ; ~ pw? (p—m) — 2” ie aa g 2 TW ae hiner ees a ey with a similar equation for 7 in terms of sin pot. The amplitude.of the vibration is a maximum when po?( pB—A) — Be? =0, or 1 eae age ei oe 2 *~ p@B=B) ae |e 9. This enables the second critical speed to be calculated without difficulty, and fig. 6 gives the necessary curves for reading off the proportional values directly. The ordinates give the values of w/c, and the abscissee the values of the ratio A/B. The method of using the curves is as follows :— 1. Work out the radius of gyration /, about the shaft centre line. 2. Work out the radius of gyration ky about a line perpen- dicular to this through the centre of gravity. . Work out the ratio of A/B or k,/k,; for turbo-generator rotors its value is usually between *2 and ‘4 and for flywheels up to 2:0. (This gives the working point on the horizontal axis of-the curve.) . Work out the “ first” critical speed in the usual way and multiply by l/h, so as to obtain the stationary “‘second ” critical (where /=half the distance between bearing centres). 5. To obtain the second critical speeds read off from the curves the figures given on the vertical axis and multiply by the stationary critical found from (4). io) is ill usually be ents represented by the bee oe a os = = 6 i g ee a 2 a ae lo) Sis, Ss og ; a oS a8 D ae Fy 5 = on a S) eo oH PS RB BS @ oo HO Ee = a Oo on o 4 o~- 1 Eio BH > + 4 & “ed cL a jo) colicin Lo z 8 Ne uae right-hand side of equations I Thus It is, however, (04) and (55). due to some external cause. 18 e that one of these components may be absent ple bl turb if there is a couple round the « axis only, equations (54) and ing cou concelva the d 1s and Critical Speeds of Rotors. 155 (55) become —By—«n+AwE=—Te’ sin pot, . . (59) Bees kam Oe viel istmer + (60) giving the forced vibration ‘ 2 9.2 Ta)” Co’ — po n= pe (es —po?)?— pat sin pot. - (61) ones mpe BEE (62) ~~ B (e2—p?w?)?— mi" p?o* LSet EN ‘ The oscillatory motion is thus a whirl the nature of which depends on the speed. The whirl is a maximum when the denominator is zero, that is, at two speeds, one on each side of the stationary critical, and given by 1 i | ole =p Em) ik There is thus a possible further second critical speed, corresponding to the + sign, lower than that already found corresponding to the — sign. For the ordinary rotating couple the direction of whirling is, of course, always in the direction of rotation, whether the speed is above or below the critical, and this is indicated by the fact that, as will be seen from equations (58), the sign of the amplitude in both planes changes, showing also that the phase of the motion has changed by 180°. But for an alternating couple about the « axis only, as will be seen from (61) and (62), the motion in the horizontal plane changes. sion at each of the critical speeds indicated by (63), while the motion in the vertical plane has a further change of sign when p’e?=c,”. It will be seen, if these changes are fol- lowed out, that the whirling is in one direction below the stationary second critical and in the opposite direction above that speed, while at the stationary second critical the motion is in the horizontal plane only, that is, at right angles to the applied torque. 11. The following example is given to illustrate the appli- cation of the above curves and formule. A rotor consists of a solid cylinder 30 ins. diameter and 60 ins. long, running in bearings, 107 ins. between centres. From the deflexion diagram, the deflexion at the centre of gravity is, say, ‘0087 in. or °0221 cm. 156 Mr. P. Cormack on Harmonic Analysis of We then have: ». The first eritical speed ¢; is |) =2020 R.P.M. Bat "148 The ratio of the radii of gyration h/t) is 6/19=0°316. Half the distance between bearing centres=53°5 ins. The radius of gyration for the cross moment of inertia ko 18°37 ins. The stationary second critical is, therefore, =0730 K.P. MM. aS D3'd i alee From the curves, fig. 6, it will be seen that the main oscillation will occur at ; 9720 x 1:21=6940 R.P.M. and that a double frequency vibration may possibly show itself at - 5730 x 0°545= 3120 R.P.M. Thus a turbo-generator designed for a speed of 3000 R.P.M. and having the above mechanical constants might show marked vibration on the overspeed test. X. Harmonie Analysis of Motion transmitted by Hooke’s Joint. By P. Cormack, A.A.C.Sel., Lingineering Dept., feyal College of Science for Ireland *. is Wie the growth of high-speed machinery, the determination of the accelerations of machine pieces becomes of increasing importance. These deter- minations are considerably simplified by expressing in the form of a Fourier Series the displacement of the piece under investigation. The value of this method in the _ analysis of the various phases of the motion of the mechanism of the direct-acting engine is well known. It is here proposed to investigate the coefficients of a Fourier Series for the angular displacement of the driven shaft of a Hooke’s Joint. The method being applicable to certain inversions of the slider crank chain, these are also included. ‘The ease with which the coefficients can be determined, and calculations made from the resulting series, make the study of these mechanisms from this aspect one of considerable interest. * Communicated by the Author. Motion transmitted by Hooke’s Joint. 157 2. In Hooke’s Joint the point B moves in the great circle CBN and the point A in the great circle CAN (fig. 1). The arc AB is constant and equal to a quadrant of the great circle. The point A receives its motion from an arm OA Fig. 1, set at right angles to the driving shaft OX, while B transmits motion to the driven shaft OY. Let the angle between the shafts be y; this will be the angle between the planes of CAN and CBN. In the spherical triangle ABC we have cos¢ = cosa cos b+sinasin b cosy, Since c=7/2, this becomes cosa@cosb+sinasinbcosy=0. . . . (1) Plainly when B is at C, A will be at T; B will therefore move through the angle a while A moves through the angle b—/2. Writing ¢ for a and @ for b—7/2, equation (1) becomes i —cos dsin 8+singdcosfcosy = 0. Put cos y=(1—n)/(1+72), and we have (1+n) cos ¢ sin @ = (1—n) sind cos 0. *, n(cos d sin 8+sin d cos @) = sin dcos 6—cos ¢$ sin 0. sin(@—@) = nsin(d+@). e(o—8) p99) — yeild +) ne-Ho+8), 158 Mr. P. Cormack on Harmonic Analysis of Multiplying both sides by ¢“?—® gives ye) 0) ata teen new"? —ne—228. eto 9) 11 — ne?]. — ee e2(o—%) = (1 ne (nee Since d—O lies between +e and —S and n» is less than unity, we may write 2i(@ —0) = log (1—ne~*) —log (1—ne?"®) : = —ne— 29 —Inre— MO _ nig 69 __ tne 4 birch 4 132618 + , Ae = n.2isin 20+ $n*.2isin4944n3.2isin60+... -. &—6 = nsin 20-57 sin 40 + an sin 60>... a enemen Tt will be evident that (2) gives the displacement of the driven shaft relatively to the driving shaft. In practice the angle between shafts joined by a, Hooke’s coupling = we have 1l+yn rarely exceeds 15°. Since cos a= a Oe For «=15° we get n=°0173, so that we can without appreciable error neglect the terms containing the square and higher powers of n in (2) and put @— 0 == nsinZ6, 0 ee a For the above value of a, the maximum value of o—O given by (3) is ‘0173 radian or nearly one degree. From (2) we have @ = 0+nsin 20+ $n’ sin 40+4n? sin60+.... . (A) a aoe 2n cos 24+ 2n* cos 46 + 2n? cos 60+ ...).(5) n = (1--cos«)/(1+cos a) = tan? 2 2 a = (5) (—4n sin 20 —8n? sin 40—12n3 sin 60—...). (6) In obtaining (5) and (6) we assume the series formed by the term-by-term derivative of the member on the right in (4) and (5) to be convergent and to converge to the differential coefficient of the member on the left. In Motion transmitted by Hooke’s Joint. 159 obtaining (6) we take the case in which the speed of the driving shaft is uniform. From (5) the maximum speed of the driven shaft is readily seen to be w{1+2n/(1—n)} or wseca; the minimum speed is seen to be w{1—2n/(1l+n)} or wcosa, wo being the angular velocity of the driving shaft. When the anole between the shafts is not lar ge, (6) may be written 8 _ansinv0(@Y ThE CTY a i a ee Thus the maximum angular acceleration of the driven shaft is very approximately 4nw’. If «=15° and w= 60 radians per sec., the maximum angular acceleration is almost 250 radians per sec. per second. Fig, 2. ee WS N \ Ys N BAR RW z 3. In the mechanism of the pe caliaanae cylinder engine, and the quick return crank and slotted-lever mechanism (fig. 2), we have sing siny_ sin (8+¢) ae QA Whe eel Chant sind = ~sin (6+6)=nsin(@+¢), where n= - 2 —e—1 = nett) — ne +9), e246 —1 = nell +29) ne 2, eb (1 —ne’”) = 1—ne-, 2 = (L—ne—) /(1—ne’*). TT ee ee ee ek omy pap! 160 Harmonic Analysis of Motion. Since ¢ lies between +5 and =37 and n<1, 2id = log (1—ne-) —log (1 — ne”) ? van: Way 0 Lr 220 SUS ho = — ne anre 4 nc 4 dnret® + ins =n.2i1sind+4n?.2i sin 204 4n? . 21 sin 30+.... o = nsin 0+4n? sin 20+ 4n' sin 30+..... . . (8) = (n.00s +n? cos 26 +n cos 30+...) “Hines (9) j 2 - = (—n sin 0—2n’? sin 20—3n? sin 30—...) (“) , 4. In the Pin and Slot mechanism (fig. 3) we have Vi = a—O—d¢.. The angle ¢ is given by (8), so that n? 2 | we = w—O—nsin p— sin 20— 3 sin SO ates ea alata) a = (1+ncos @+n? cos 20 +n cos 30+...),(12) 2 2 ju a (Fi ) (n sin @-+ 2n? sin 26-+ Bn? sin 36-+-.,.). (13) It will generally be found that we need to consider but the first few terms of these series in making numerical calculations. Lee AGRE 4 XI. Short Electric Waves obtained by Valves. By E. W. B. Gi, W.A., B.Sc., Fellow of Merton College, Oaford, and J. H. Morreut, M.A., Magdalen College, Oxford *. x. HERE have recently been discovered methods for the generation of continuous oscillations of short wave length (of the order of aboat a metre) by means of three eect: ode valves. In January 1920, Barkhausen and Kurz + found that with hard valves—. e., valves at extremely low pressure, if the filament and the plate were approximately the same potential, or, indeed, if the plate were at a potential considerably lower than the filament, provided that the grid was. kept at a high potential with regard to them, continuous oscillations could be maintained in a cirenit of the Lecher Wire type connected to the grid and plate. The wave length depended primarily on the grid voltage, but also on tlie emission from the filament and on the plate voltae. Whiddington f had previously described another method of getting oscillations of lower frequencies using a soft valve, 1.é@., a valve containing gas at low but appreciable pressure. He employed more usual circuits for a valve, in that the plate was at a high positive potential with regard to the filament and the orid at a few volts above the filament. In this case longer waves were emitted, and he noticed that if V was the orid potential and »X the wave-length emitted, then >7V was constant §. There appear to be other arrangements not hitherto re- corded which will also give these waves. With a hard valve and with the grid at a positive potential, oscillations can be. obtained if the Lecher Wire system is connected across the filament and grid; the plate may be positive, negative, or at . the same potential as the filament, or it may be insulated. Further, the third electrode—the plate—is unnecessary, for aalincione can be sustained by means of a valve consisting of a filament and an anode formed as a spiral of wire con- centric with the filament, when these two are connected to the Lecher wires. An intermediate arrangement has been worked successfully in which the wave- length of the diode connected as above is modified by a cylinder concentric with * Communicated by Prof. J. S. Townsend, F.R.S. + Physikalischer Zeitschrift, Jan. 1920. {t Whiddington, * Radio Review, Nov. 1919. § For a general account of these experiments see ‘ Radio Review,’ June 1920. Phil. Mag. S. 6. Vol. 44. No. 259. July 1922, M oe nail erence, high Ps Xue TT Ee eS 162 Messrs. E. W. B. Gill and J. H. Morrell on Short the anode, but placed outside the valve and set at various potentials. The best conditions for these cases are still under investigation. 2. Barkhausen and Kurz were apparently unable to give any explanation of the way in which the oscillations were sustained, while Whiddington assumed that the emission of ions from the filament was discontinuous and occurred in bursts. The authors, on the other hand, do not think that any special assumptions are necessary, and that the ordinary conditions for the maintenance of oscillations by continuous emission will account for all the tacts they have observed, provided that the time taken by the electrons to pass between the electrodes is taken into consideration, as this time is of the same order as the period of the short waves. In the present paper only oscillations of the Barkhausen type are considered in detail, but the theory can be extended to cover all the types, and an account of some experiments on the last type (with a diode) will be published later. It is worth noting that certain writers give the impression that the seat of the oscillations is in the gas or in the electrons in the valve, and that the Lecher wires connected to the valve serve only to demonstrate their existence*. It appears from our experiments that the wires or conductors attached to the electrodes are a necessary part of the oscillatory system. Hven with the Lecher wires removed, there will always be some circuit composed of the connecting wires to the batteries or even the valve leads up from the sockets, which will have natural periods of a suitable order for short wave oscillation. This fact seems to have been overlooked in some recent determinations of ionizing poten- tials, where large emissions from a heated filament were used as a source of electrons. Oscillatio:s will take place even when the valves contain a small amount of gas, but in all the experiments described in this paper gas-free valves were used. 3. It will probably be most convenient first to describe the experiments in detail, and then to set out the theory and apply it to the observed facts. Various valves were used, but mostly the Marconi M.T.5 valves, which were very kindly given to us by the Marconi Company. These valves consist of a straight filament FF held in the centre of the valve by springy arms. The ad- vantage of the spring is that when the filament is heated and * Whiddington’s theory is independent of there being any external tuned circuit. ao. aia 524 —— : Electric Waves obtained by Valves. 163 expands, the spring prevents sagging. Surrounding the filament is a cylindrical wire grid, GG, composed of thin wire of square mesh, each square having a side of about 15mm. The lead to the grid goes out at the bottom near the filament leads. A cylindrical plate, PP, surrounds the whole with its lead going out through the top of the bulb. These valves being used for transmitting purposes are very thoroughly “glowed out”? and pumped tas avery high vacuum. The filament emission is very high when heated w with 6 volts direct, and for the low emissions that were generally used it was very constant. As the plate lead passes through the top of the bulb, instead of through the bottom and the sealed socket, very high insulation is obtained, and, if a strip of tinfoil ‘connected to earth is placed cama the outside of the glass, very small anode currents may be measured by an dieeiiombice without any disturbance due to leakage. It is not necessary for ordinary wireless purposes that the valves should be constructed with the grid and plates either accurately circular in section or accurately centred with regard to the filament ; but for the purpose of calculation a symmetrical system of electrodes is necessary and the M.T.5 valve used in most of the experiments was specially selected. All the numerical results to be quoted were obtained from this valve. There is no difficulty in getting the slort-wave oscillations with many types of hard valve, Whe French type produces them quite easily, but the chief reason for selecting the Marconi M.T.5 type was that the electrostatic field between the square-mesh grid and the plate approximates M 2 164 Messrs. BE. W. B. Gill and J. H. Morrell on Short much more closely to the calculable field between two co- axial cylinders than does the Held in the French type, where the grid is a spiral coil of fine wire. The diameter of grid - used was 1 em. and that of the plate was 2°5 cm. to an accuracy of about 5 per cent. | 4. The preliminary experiments were made with the apparatus arranged as in fig. 2. The valve is shown dia- grammatically: F is the filament, G the grid, P the plate, LL the Lecher wires, which were of copper wire each about 850 cm. long and spaced 5 cm. apart. ‘They were suspended about 200 em. above the floor from insulators secured to the walls at each end, and from one end were leads about 70 cm. long to the grid and plate respectively. The bridze consisted of two equal condensers, ©, C, joined through the heater-coil of a Paul thermo-junction, T. The outer plates were fitted | with contacts to slide along the Lecher wires. The capacity of these condensers is unimportant, provided it is large com- pared with the capacity of the valve. In practice, the capacities were of the order of 1 milli-microfarad. The terminals of the thermo-junction were connected to a gal- vanometer by two long leads, which are not shown. The sliding contacts were also connected to the negative side of the filament-heating battery B, that on the grid-wire through a high-tension battery V,and that on the plate-wire through a potentiometer S. which could raise the potential of the plate +6 volts above the negative end of the filament. Two sensitive milliammeters, A, A, gave the steady currents through the valve to the grid and filament respectively. A rheostat, R, controlled the filament-heating. In all cases potentials are measured with regard to the negative end of the filament. With this arrangement the electrons set free at the Electric Waves obtained by Valves.: 165 filament move outwards under the positive voltage, V, of the grid, and a certain number go direct to the grid and are collected there, the remainder pass through the grid, and, if the potential of the plate is just less than that of the filament, they return to, and are finally collected on the grid. If, on the other hand. the plate potential is a little above that of the filament, a certain proportion of those getting through the grid reach the plate. If the plate potentiometer is now ad- justed till the plate current is just zero, and the bridge is moved along the wires, it will be found that with the bridge in certain regions a plate current appears. It was the appearance of this plate current which led Barkhausen to the discovery of the short waves. With the present apparatus these oscillations are also made apparent by the deflexion of the galvanometer attached to the thermo-junction. The positions of the bridge at which the galvanometer gave a maximum deflexion were fairly sharply defined, and did. not always coincide with the positions for maximum plate current. It is not necessary for the plate potential to be so adjusted that the plate current is just zero when oscillations are not occurring. ‘The plate may be set at a considerable negative potential, or the plate voltage may be positive. It was found that for 'a given grid potential there isa certain plate potential at which the oscillating current through the thermo-junction is a maximum. Also as the potential of the plate was increased, for plate potentials only slightly positive, if oscil- lations commence the plate current increases ; at a certain plate potential no change is noticed in the plate current ; and at higher potentials the plate current decreases. For the M.T.5 valve this critical potential was about +2 volts, when the voltage drop down the filament due to the heating current was about 4 volts. In the first experiments with this apparatus the position of the bridge was varied and the current in the thermo-couple observed when the grid voltage V, the heating current, and the plate potential were all kept constant. The oscillating circuit consists of a condenser formed by the plate and grid of the valve, the distributed inductance and capacity of the Lecher wires up to the bridge, and the capacities C, C in series with the wires and with the short resistance of the thermal heater which connects them. Hence, if there is an optimum wave-length A corresponding to the grid voltage V, and if, starting near the valve, the bridge is pushed along the wires, maximum amplitude of oscillation eS ee 466 Messrs. E. W-B. Gill and J. H: Mortell-on Short should occur when the above circuit is tuned to A, 2A, 3r etc., these positions being indicated by the deflexions of the galvanometer connected to the thermo-junction. Moreover, the distances measured along the wires between successive positions of maximum oscillations should be equal to S and all therefore should be equal. It was soon found that this simplicity was not attained, in certain cases equi-spaced positions were found, but in the majority of cases there were millet two sets of positions forming two series of equal spaces, which, as the spacing distance of the two sets was different, appeared to indicate two optimum wave-lengths. These effects are due to the different modes of oscillation of the system, and, according to the theory which we give below, a grid voltage V will, under suitable conditions, sustain oscillations of short wave-length between certain limits. Any mode of oscillation corresponding to a wave-length between these limits will be maintained. It was therefore desirable to arrange the apparatus so as to avoid these complications. 5. The most obvious improvement was to give up the idea of finding the wave-lengths by moving the bridge, and to put the br idge and its leads at the far end of the parallel wires joined to the valve, and to measure the wave-lengths of the oscillations by means of a loosely coupled secondary circuit. The system of wires connected to the valve is thus fixed. A second pair of long Lecher wires were set up with a loop joining one end, and this loop was brought near the valve circuit. When the secondary is in tune with an oscillation in the primary the current in the primary is reduced. The deflexion cf the galvanometer connected to the thermo- junction in the primary cireuit may be reduced by 50 per cent. when the bridge in the secondary circuit is in the tuned position, and a movement of 0:5 em. either way will restore the deflexion to its original value. The distances between the successive positions of the bridge on the secondary circuit, for which the deflexions of the galvanometer attached to the primary circuit are a minimum, are the same, and are equal to half the wave-length of the oscillation in. the primary circuit. All the wave-lengths quoted were measured on this form of wave-meter and may be taken as accurate to 0°5 per cent. * W:th the condenser bridge and thermo-couple at the far end of the Lecher wires the filament was heated to give an emission of a few milliamperes (this is low heating for an * Townsend and Morrell, Phil. Mag. Aug. 1921, pp. 266-268. Electric Waves obtained by Valves, 167 M.T.5 valve) and the grid voltage was raised by two volts at a time by means of batteries of small accumulators from 16 volts to 120 volts, while the plate was kept about 2 volts positive, as this gave large deflexions. The corresponding galvanometer deflexions are shown in fig. 3. ! The deflexions are plotted against grid volts ; as a thermal detector was being used, the deflexions are proportional to the mean square of the oscillating current. The curve shows that oscillations are occurring over nearly the whole range, but that there are maxima for certain voltages—viz., 16, 24, 42, 58, 82, 114, approximately. ‘The wave-lengths measured as above give from 16 to just below 24 volts 1586 cm., from 24 to 40 volts 1451, and so on, the wave-lengths for successive portions of the curve being 366, 307, 262, 233. These correspond to the free oscillations of Fig. 5. 82 Vof. the system, the wires of which were 850 cm. long with leads to the valve about 70 cm. long, with a slight addition for the leads within the valve itself. The system of wires connected to the valve therefore present a selection of various modes of oscillation with wave- lengths 586, 451, 366, 307, etc., cm., from which the valve chooses the one suitable for the particular voltage V between the grid and plate—the sharp rises just before the various maxima showing that the system oscillates on the longer wave-lengths by preference. For each particular wave- length there is a certain grid voltage which gives the strongest oscillations when the heating current in the filament and the plate voltage is constant ; but the heating current and the potential of the plate relative to the filament both affect the optimum voltage for a given wave-length. In- creased emission has the same effect, but this effect depends on the degree of saturation of the emission current. In the preceding experiments the wave-lengths of the oscillations were measured with a constant heating current 168 Messrs. E. W.B. Gill and J.H. Morrell on Short — in the filament, but the current from the filament varied with the grid voltage. For the lower voltages all the electrons leaving the filament do not reach the ‘grid space, some returning to the filament. or theoretical reasons it 1s more convenient to find the grid voltages which give the maximum amplitudes of oscillation on the various wave- lengths when the heating current is so adjusted that the same current flows from the filameut to the grid space for all the voltages, the plate voltage being kept constant as before. The table below gives a set of experiments done under such conditions with an emission current of 6 milliamperes, and the plate at 1:3 volts positive to the filament. In column 1 are given the wave-lengths Xin cms., in column 2 the grid volts V, which excite these wave-lengths most strongly, ‘and in oldie d the product A?V :— Xr. Vv. rv. 208 em. 156°5 68 x 10° 233 L225 66 A 262 92°5 64 307 68°5 ” 64°5 366 50°5 67:5 451 36°5 74 All these results, with the exception of the last, agree well with the relation \?V = const. It is not difficult to see whv this agreement should be less exact as V decreases. The electrons concerned are not all moving under similar conditions. Owing to the voltage drop of the heating current down the filament, the field between filament and grid differs by about 4 volts for electrons starting from the extreme ends of the filament. And when V be- comes comparable to this 4 volts a disturbing factor is introduced. 6. These experiments thus give the grid voltages which produce the strongest oaurlle Wome on coria donnie wave- lengths determined by the particular length of wire used. To find the range of wave-lengths fieiniaiaed by a given erid voltage a slightly different apparatus (fig. 4) was “used. An adjustable circuit was constructed of two rods, and two telescopic tubes fitted over the rods, so that the effective lengths of the system could be varied by sliding the tubes over the rods. The condensers and thermo-junctions were attached at the ends X, X! of the rods, and the ends Y, Y?! of the tubes were connected to the plate and grid of the valve respectively, the other connexions being as before. Electric Waves obtained by Valves. 169 For brevity, the adjustable circuit will be referred to as the rods. It is not possible to graduate the rods in wave- lengths as against extension of the arms, as this wave-length depends on the emission and on the plate volts. Thus with a fixed length of the arms and 44 volts between grid and plate: (1) With plate potential fixed. Emission 2°2 m.a. XN=311 cm. 6°8 306 9°38 300° (2) With emission constant at 5°2 m.a. Plate potential 1°2 volts. A=O0S em: 2°4 314 - Hence for a given setting A decreases as the emission rises, and increases as the plate voltage is increased. This is due to the fact that the plate and grid are not a potential node of the oscillating system, but are a variable distance from it depending on the alternating voltage necessary to sustain the oscillation, and this in turn depends on the emission and plate voltage. It is vot, however, necessary to go further into this, as the wave-lengths were always found directly by a secondary circuit as in Paragraph 6, the rods being used as 2 convenient way of varying continuously the wave-length of the system connected to the valve. With all the other factors fixed, the rods were. pulled out a centimetre at a time and the oscillating current and wave-length recorded for each position. In one experiment the emission was 1°5 m.a.. the grid potential 44 volts, the plate potential 1°8 volts, and oscillations were maintained from }=320 em. to X=451 em. with a maximum oscillation about X= 323cm. It was always found that the maximum oscillation was close to the short-wave end of the range. 170 Messrs. ©. W. B. Gill and J. H. Morrell on Short The effect of (A) varying the emission current keeping the grid and plate voltages constant, and (B) varying the plate voltage keeping the emission current and potential between grid and plate constant was investigated with this apparatus. In (A) increased emission broadened the range and de- creased the wave-length of maximum oscillation. For example, with V,,=44 volts and V,,=1:2 volts the wave-length for best oscillation with total emission 7:0 m.a. was 295 cm. With total emission 10°6 m.a. it was 274 em. In (B) increase of plate voltage increased the wave-length and also broadened the range. Thus with V,,=44 volts, and total emission 3°8 m.a. with V,,-=1°2 volts X=298 cm., © and for V,r=3°0 volts X=3821 cm. This last observation | must not be confused with the case in which the potential of the plate is increased and that of the grid kept constant. In that case also, increase of plate potential increases the length of the strongest wave, as was observed by Barkhausen, who attributed all the difference in wave-length to the alteration in potential difference between plate and grid. This cannot be the whole of the explanation, for, as stated above, similar results may be obtained by raising both plate and grid equally with respect to the filament. 7. A simple theory to account for the maintenance of the oscillations can be worked out by making some simplifying assumptions ; but a general theory will not be attempted, partly because the resistance of the oscillating circuits used was unknown and partly because if the assumptions are not — made the calculations become extremely complicated. These assumptions are :— A. That the grid and plate ean be regarded as forming a parallel plate condenser. — - B. That, of the electrons which leave the filament, a fixed small proportion pass through the grid in a uniform stream, and that each electron on passing through the orid has the same velocity. C. That the electrons which return to the grid from the plate side are nearly all collected directly on it, 7. e., only a few pass through on the return journey. D. That the oscillating potential differences are small com- pared with the fixed potential differences employed. It is also assumed that the pressure of the gas inside the valve is so low that the number of collisions between electrons and gas molecules is negligible—this is certainly true for the valves used. Klectric Waves obtained by Valves. Lea With these assumptions we shall only attempt to show that an oscillation can be maintained of about the right order of wave-length. I'he principle involved is the fellowing :— Suppose the filament and plate are at zero potential and the grid at + V ; then the electrons from the filament which pass “thr ough the erid with a velocity v due to the potential V come to rest at the surface of the plate and return to the grid, which they again reach with velocity v. In the space between the erid and the plate the total work done by the fixed potential V on the electrons which move in this space is zero, all the work having been done between the filament and grid. If now superposed on the fixed potentials there is an alternating potential V, sin pt between grid and plate due to oscillations, the work done by the potential V)sin pt on the electrons is not necessarily zero. If the work is positive the electrons are abstracting energy from the oscillating system, and the average velocity with which the electrons hit the grid is increased; the oscillations cannot in this case be sustained by the movement of the electrons. But if the work is negative the electrons are giving energy to the oscillating system, and if the rate at which this energy is given is at least equal to the rate of dissipation of energy in the oscil- latory circuit by resistance, radiation, or dielectric loss, the valve will maintain the oscillations. The average velocity with which the electrons hit the grid is in this case less than the velocity v due to the potential V which they acquire between filament and grid, and hence the energy put into the system from the battery V is not all used in heating the grid but part is turned into energy of oscillations. The above argument is not affected if, in consequence of the oscillation, some of the electrons are collected on the plate. In all cases, provided the total work dune per oscilla- tion by the alternating field is negative, an oscillation can be sustained if the dissipation of energy in the oscillatory circuit is not large. 8. The particular ease in which the filament and the plate are at the same potential when there are no oscillations may be considered first. Let V be the potential above the plate of the grid and d the distance between them. When there are no oscillations the electron passes the grid Wy with velocity v |= py a ne v] and is then subject to a constant m retardation f, which brings it to rest just at the plate. If T 172 Messrs. E. W. B. Gill and J. H. Morrell on Short is the time the electron takes to pass from grid to plate, v=fT. A further interval of time T brings the electron back to the grid with velocity v. Assume now that superposed on the fixed potentials is an alternating potential Vosin pt between plate and grid; the electric force due to this in the space between plate and grid Tr is qin pt, and if —eisthe charge onan electron the corre- Pay tect 4 HON ane sponding force on it is —7 sin pt towards the plate. Since Vy is taken to be very:small compared with V the motion of the electron may to a first approximation be taken as determined solely by V, 7.¢., its time across is T and retardation f. | | The work done by Vosin pt depends upon the time fp at which the electron passes the grid, and for a particular value | deV o of t the work is equal to 7 sin ptdx. The axis of # 0 being perpendicular to plate and grid and #=0 being on the atc But the velocity at time ¢ is ada “ Te Od Mi Oat Need eto) and the above work reduees to ott fe “ (T+ to—t) sin pt dt, to which finally gives : Work on electron going from grid to plate = 2eVo/T cos pto sin pto—sinp Tt +.ty\_ l ey p di i py? SRE L Oe (1) similarly, the work done on the same electron as it returns from plate to grid comes out as 2eVo ak aE PPR TN “pe (cos to + 2'1 aaa sn pt,+T—sin pto+2T A ; a) ), (2) ; P Thus the velocities of the electrons on their arrival at the plate or on their return to the grid depends on ¢o, that is, on the value of Vosin pt at the instant they pass through the grid. Assuming a constant stream of electrons through the grid, it is easily seen by integrating (1) for values of between O and = that the total work done per period is 0 Electric Waves obtained by Valves. Rae SC and similarly for (2). Hence, if all the electrons returned to the grid an oscillation would not be maintained. The possibility of a maintained oscillation depends in this case on the fact that in each oscillation a certain group of the electrons are collected on the plate and the integral of (2) does not in consequence include all the values of t) between 2 2Qar 0 and — and its value is not therefore zero, but may be Pp eee, negative. The first step is, ietctore: to find which electrons reach the plate. When there are no oscillations the electrons have sufficient energy on passing the grid to just take them to the plate against ‘the potential V, uel if therefore any extra work is done on them they will be collected on the plate, but if the work is negative they will fall short of the plate and return to the grid. Expression (1) shows that ali the electrons which pass through the grid at times ¢), such that T cos pty if sin ptp—sin p(T + to) — ) 2 plate, while those for which it is negative just fail to reach the plate and return to the grid. OF the electrons then which pass the grid hale go on to the plate and half return to the grid, the electrons. running to the plate for a time equal to |p (half the periodic din of the oscillation) and then running back to the grid for time z/p and so on. But the total work done “by the oseillating potential on the two halves as they go from grid to plate is zero; and therefore the net work done is the work done on the return journey on the half which returns to the grid. To find therefore if an oscillation whose periodic time is 277/p and amplitude V, can be sustained by a grid voltage V it is necessary first to find the time T w hich the electrons take to pass from the grid to the plate under the field due to V alone, next to find. from equation (1) the values of ¢, for those electrons which return to the grid when the system is oscillating, and, finally, by taking the mean value of expression (2) \ for fies values of fo and, knowing the emission current, to find the total work done per second by the oscillating potential. If this work is negative and at least equal to “the dissipation loss per second, the oscillation will be maintained. A table of calculated approximate results is given below for various values of the ratio T:1/p.. The second column gives the values of pt) for the electrons which return. is positive, will reach the 174 .Messrs. E. W. B. Gill and J. H. Morrell on Short to the grid in each oscillation, the third gives the total work done per second by the oscillating potential in arbitrary units for a fixed value of V, and of emission, and the fourth gives the corresponding wave-lengths for the particular value, V=44, for which T==4°3 x 107° second (see next paragraph) :— | Values of pty for electrons which Work. Vo return to grid. 165° to 345° negligible. 1040 cm. ny Do AT Tw = 50° ,, 330° —" 200 P= oy 150° ,, 330 47 5 pee 135° ,, 315° —-85 Suge ay B= ap ve TOe | e00R ~"36 260, ad © ,, 180° 32 130 o=e 90° ,, 1 oe In all these cases the work is negative and oscillations can be theoretically sustained, though in practice the dissipation losses in the oscillatory circuits are such that generally only those wave-lengths corresponding to the larger values of the work exist. For all values of the ratio T:1/p outside the range of the table the work is small. The larger the work the greater the amplitude of oscillation that will be sustained, and the periodic time 27/p of the oscillation of maximum amplitude for a given value of ‘I’ is seen to be in the region of 8T. In general, the wave-length corresponding to this will be sustained, and aiso a certain range of wave-lengths ~ on both sides of it, the limits of the range being determined by the dissipation losses in the. oscillatory circuit. The theoretical result is in good agreement with several of the experimental results of paragraph 6. In the particular case recorded there for V=44 the range of wave-lengths sustained was from 320 to 451 em., with a maximum amplitude for 323 cm. Increase of emission broadened the range of wave-lengths sustained, which is in accordance with the fact that for a given Vo the work put into the oscillatory system is proportional to the emission current. The fact that the wave-length of maximum amplitude of oscillation was near to the short wave-length end of the range also agrees with the calculated fact that the Electric Waves obtained by Valves. 175 work done falls off much more rapidly on the short-wave side of the maximum than on the long-wave side. 9. To calculate the time T an electron from the filament takes to go from the grid to the plate when the grid potential is V volts above both filament and plate it 1s not necessary to assume the grid and plate to be parallel, but they may be taken, as they actually are, to be concentric cylinders of radii a, 0. The retarding force on the electron when it is at distance v from the axis is 2 where k=V/log.~ The equation of motion is therefore : d*r —ek i ae AS dt? yr? which gives when integrated twice, remembering that dr i 0, when r= 0, bE ee a ey gee ae ee (ae af oda eV Seo i é aN. In the actual valve used a=*5 cm., b=1'25 cm., and, taking é = 7 _ . e “+= 5:3 x 10" E.S. units and measuring V in volts, qe 200 x 107° V/V the accuracy of this being limited by the accuracy to which a and 6 are known and probably from 5 to 10 per cent. The wave-length for any relation between T and 1/p can now be at once calculated. If p=nz/T, the time of one oscillation is 2a/p or 2T/n and the wave-length in cm. is 6x 10° T/n. oie The simple theory shows that for the oscillations of maximum amplitude pT has a certain value about 37/4. But second, 1 1 T x Fs andrA« a and hence the connexion between the Vv grid voltage V and the wave-length » of maximum oscillation is A7V =constant. 10. The theory is thus in good general agreement with the experimental results, but there is one fact unaccounted for—that being the variation in the wave-length of the 176 Messrs. BE. W. B. Gill and J. H. Morrell on Short oscillation of maximum amplitude, for a fixed potential between grid and plate, when either the emission is altered or the plate potential is slightly altered with respect to the filament. There is also a special case, which is forming the subject of a separate investigation, in which, when ‘the plate is very negative (40 volts or so} with respect to the filament, oscillations can still be produced, but without any current reaching the plate at all. These oscillations are, however, _ very much weaker and more difficult to produce than those dealt with in our experiments. The simple theory which depends on the collection on the plate in each oscillation of a group of electrons will ob not account for this special case. The explanation of the above considerations is to be looked for in the assumptions made in the simple theory. The first assumption that the grid and plate could be regarded as parallel is not important, as the field between cylinders of the size of the grid and plate used is not far from uniform. (It will be noted that the value of T was calculated for the valve used by taking the field between cylinders.) The only difference between cylinders and parallel plates on the simple theory would be to make the ratio T to 1/p for maximum oscillation slightly different. But the second assumption that the electrons pass the grid in a constant stream all having the same velocity requires more careful examination. This velocity is not actually the same for two reasons : A. Because there will be alternating potentials between the filainent and the grid which set up. a velocity distribution at the orid. B. Because of the voltage drop of the heating current down the filament. In the usual methods of producing oscillations by means of valves the alternating potential of (A) is most important, as it controls the whole action of the valve, but in our experiments it is only of secondary importance. The alternating potentials induced between filament and grid are smaller than those between erid and plate, and the major part of the work done by “the alternating field on the electrons, which is what determines whether the electrons reach the plate or not, is done between grid and plate, and it is therefore nearly correct to say that all the electrons passing the grid at times ¢, such that expression (1) is positive, reach the plate. In the extreme case, however, when the plate is. so Electric Waves obtained by Valves, 177 negative that the system is unable to oscillate by the method of driving groups of electrons on to the plate, the oscillation is almost certainly due to a velocity distribution at the erid, as this means that the electrons do not pass the grid in a uniform stream, and allows the integral of expression (2) to be finite and not zero, as it normally is when all the electrons return to the grid. The comparative weakness of the oscillations in this case shows that the electrons have all nearly the same velocity when passing the grid. The simple theory should therefore be in agreement with the observed facts, as it is when the oscillations are mainly due to the collection by the plate, but as the plate is made more negative with respect to the filament the velocity dis- tribution at the grid becomes more important and the simple theory is less accurate. The velocity distribution at the grid will also bo affected by the emission, as this varies the space charge round the filament—this affecting the time the electrons take to pass from the filament to grid,—and this in turn varies the small effect of the alternating field in this space. The effect of the voltage drop of about 4 volts down the filament is that, instead of dealing with one stream in the field due to the grid being charged to V volts, there are a series of streams moving under potentials varying from V to V—4 (V being the potential difference between the grid and the negative end of the flament). The number of electrons in the various streams varies from a maximum number corresponding to V—2, the middle of the filament being the hottest. The emission falls off equally on both sides of this middle point. In the general case, when the plate is slightly positive with regard to the negative end of the filament when there are no oscillations, some of the streams reach the plate and the remainder approach it closely, but to varying distances. IE oscillations commence some of these latter streams are periodically diverted to the plate, while in the other half oscillations some of the former are diverted off. Thus all the streams concerned maintain the oscillation as in the simple theory, and unless V is small the wave-lengths they each maintain best are nearly the same, so that the combined effect ditfers little from that of a single stream moving under potential V. The ¢ question of whether the mean plate current rises or falls when oscillations begin depends on whether the average Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. N 178 Short Electric Waves obtained by Valves. density of the streams diverted to the plate exceeds or is less than that of those diverted from it. If the plate is at the same potential as the centre of the filament, no, change should therefore occur. If it is above this the current should drop, and if below the current should rise, which is in agreement with the results in paragraph 4. In conclusion, we should like to express our thanks to Professor Townsend, who has assisted us with much valuable advice and criticism. Note on the Determination of Ionizing Potentials. In the experiments above described, the electric fields in the spaces between the grid, filament, and plate are similar to those used in experiments on the determination of critical potentials when a small quantity of gas is introduced. In the latter experiments variations in the plate current are observed as the grid potential is raised, and at certain potentials of the grid abnormal variations in the plate current are observed which are interpreted as indicating certain critical potentials, characteristic of the molecules of the as. : The experiments which are here described show that abnormal variations in the plate current are to be expected, due to oscillations which may be maintained where large currents of the order of a milliampere flow from the filament towards the grid when the gas is ata very low pressure. The effect of varying the potential of the grid is clearly shown by the curve of fig. 3. In all the ordinary methods of wiring the valve to the cells and galvanometers or electrometers, the system seems to be as often in a state of oscillation as not. Hven if the plate is so negative that the oscillations are of the weaker type which do not affect the plate current, the difficulty arises that the oscillations superimpose an unknown potential difference between filament and grid, and the electrons are not moving under the fixed field alone. It is necessary therefore, in order to obtain reliable values of the critical potential, to take precautions to prevent oscillation, which is best done by using emission currents much less than a milliampere. pesk?g:.-] XII. Pseudo-Regular Precession. By Sir G. GREENHILL*. a. is the gyroscopic motion described and illustrated in Klein-Sommerfeld’s Kreisel- Theorie, p. 209, where a spinnving top appears at first sight to be moving steadily in uniform precession at a constant angle with the vertical, but on closer scrutiny the axle is seen to be describing a crinkled curve of small loops or waves; so that in this penultimate state a realisation is obtained of a motion expressible by a function which does not possess a differential coefticient, a paradox fascinating to a certain school of pure mathe- maticians, A pseudo-regular precession, although invisible, would not feel impalpable to the analytical thumb passed over it, which would detect a roughness. But in regular precession the curve would feel quite smooth. In the investigation the axle may first be supposed moving in perfect steadiness with no trepidation or nutation ; and then to receive a small impulse, blow or couple, giving rise to the pseudo-regular precessiun visible to the eye. 1. Begin witha rapid spin about the axle, held fixed at a constant inclination, taken at first as horizontal for simplicity, in fig. 1. Bie. 1. For visible experimental illustration, it is convenient to take a large (52-inch) bicycle wheel, mounted on a spindle with ball bearings, and to prolong the spindle by screwing on a stalk, at one end or both. The end of the stalk may be supported on the hand and the wheel set in rotation by a swirl; the hand accompanies the wheel in the precession ; or else the point may be placed in a cup fixed on the floor. When the axle is released from rest, it will start from a cusp and sink down, then rise up again in a regular series of loops or festoons ; so that to secure the uniform precession * Communicated by the Author. N 2 : 3 180 Sir G. Greenhill on an impulse couple must be applied, given by a horizontal tap of appropriate amount. The word moment or momentum is of such frequent occurrence in dynamical theory that we prefer to replace it by zmpulse instead of momentum, linear or rotational. Representing then the impulse (rotational) CR due to the rotation R about the axle by the vector OC, the impulse required to start the rotation from rest, or reversed to stop it again, the axle OC in steady motion will move round the vertical OG at a constant (horizontal) inclination, with pre- cession mw, such that the vector velocity of K, the end of the resultant impulse OK, is equal to the impressed couple of gravity ; then CRu= gMh, MA denoting the preponderance or first momeni about O. This result, true accurately when the axle OC is _hori- zontal (fig. 1), is obtained at any other inclination @ of the axle with the vertical (nadir or zenith, figs. 2, 3) in the elementary Kindergarten treatment, where the top is supposed spinning so fast that the deviation is insensible of axial impulse OC and resultant impulse OK, and then the velocity of C may be equated to the gravity couple, making CRw sin @=gMh sin @, as before, when the axle was horizontal, on dividing out sin @. 2. Hanging down inert, vertically from O in fig. 2, the top forms a compound pendulum of S8.H.P.L. (simple-equivalent- pendulum-length) OP=l/=A/Mh, A denoting. the trans- verse-diametral M.I. (moment of inertia) at O, and © as above the axial M.I.; and in small invisible oscillation in a plane, the top will swing as a pendulum, and beat n/m times a second, where n?=g/l, An?=gMA, or make a swing or beat in min seconds ; Mh may be called the Regen ne about O. Falling down from rest from the upward vertical position, the to» will have acquired the angular velocity @ in the lowest position, such that by the Energy-Principle, 4 Aw?=2qMA, and the equivalent rotational impulse Aw=2An, while An?=gMh is the equivalent of P in the Krezsel-Theore. The impulse 2An=2/(gMAA) is a dynamical constant of the top, and to a geometrical scale may be represented by a length &, in addition to the 8.H.P.L. OP=/ measured along OC; and then if in any assigned top motion the constant impulse component G and CR, about the vertical OG and Pseudo-Regular Precession. 181 the axle OC, is represented to the same scale by 0 and 0’, 0; 0! ie G, CR TA h, h! RSS ODA eos 8 in Darboux’s notation (a different use of hk from that employed above). Time can be reckoned in the pendulum beat, m/n seconds ; and the relation, CRu=gMh=An?, can be written He An EB or expressed in words, the number of beats per circuit of the axle is C/A times the number of revolutions of the top per double beat. The resultant impulse vector being OK, the component perpendicular to the axle, if horizontal, as in fig. 1, is gMhA _ A?n? CB er COR? or to the geometrical scale, OC. CK=i4?, in the steady, regular precession. For brevity we are allowed to assume tacitly the geo- metrical scale, and to replace any dynamical quantity in an equation by its vector length, such as the axial impulse CR by the vector length OC, or 0’. 3. To change this steady motion of the. axle into a penultimate pseudo-regular precession, another impulse is applied about a vertical axis, supplied by a hase ts tap on the axle perpendicular to the plane OCK, in fig. 1 This will cause CK to grow to CKs, and the resultant ‘impulse to change from OK to OK,;; and to make the pseudo-regular precession advance through a series of cusps, we find that KK;,=CK, and the axle rises from OC to OC, at an angle @, with the upward vertical, zenith ; where (, reaches the level GK; of K,; and here 63, the inclination in the lowest position, is $7. By a general dynamical principle OK,?—OK,?=4k? (cos 8; —cos 63) =20C. CK (cos 6,—cos 63). ee For in the general unsteady motion of the axle of a top, where the inclination @ is varying, a new component KH is added to the impulse OK perpendicular to the vertical plane 182 Sir G. Greenhill on GOU, of magnitude Ane and then the resultant impulse vector OH describes a curve in a horizontal plane GHK, with velocity equal to the gravily couple gMhsin 0. Fig. 2. : The radial velocity of H in the horizontal plane is then j : eae = gMh sin 0 cos GHK = gMhsin ae | dGH £7 eae GH aie == gMhA sin 0 AE 5 : and integrating, 4 GH? =g9MhA(E — cos @) = A’n?( EK —cos 8) | =ik (H—cosé), 40H? =i (F—cos9), | to a geometrical scale, with HK, F' dynamical constants. : The component HK is zero in the upper and lower position, where = (65, 6s. In the general unsteady motion of the top, the impulse vector moves from OK to OH with KH=A% and with 2 KH*= OH? - OK’=OH? — ae 2 Y az Uh annie ebapyes Sa ee OU cont ee dt sin? 0 2 Pseudo- Regular Precession. 183 or with cos @=z, and replacing OG, OC for dynamical homogeneity by Darboux’s 2A, 2AN’, (S) = 2n®(F—2)(1— 22) —4(? — 20M cos 0+ W'2) = 2n2Z, thus defining z as an elliptic function of ¢, Resolved into factors, we write L=21— 2% .29—2 2 — 23, in the sequence © >2;>1>2.>z2>¢3> —1; and then z=c,sn*imt+e,cn?bmt, m?=2n?(2;—23). 4, Here with OC horizontal in fig. 1, cos 6;=0, OK,?— OC,?=G;3K,?— G,K,?=CK,? —2Z9UC 20 Weast,—20K..CK, | CK.=29CK. A greater impulse would make the cusps open out into loops in the pseudo-regular precession; but the cusps would be blunted into waves if the impulse was reduced. Reverse this tap, and K is brought back again to C, and the axle would fall as at first from a cusp and rise again. In the first cusp motion where the axle rises to a series of cusps and sinks again to the horizontal, the motion is found to be pseudo-elliptic and can be expressed in a finite form, sin 8 exp (w—ht) 2 = V (1—cos 8, cos 8) +2/(cos 4 cos d—cos? 8), connecting azimuth yr with @, the inclination to the zenith. The verification is left as an exercise. Here h=h’ cos 0,. In the second cusp motion, where the axle is horizontal and falls from a cusp, and then sinks down to an angle @; with the downward vertical, nadir, the (4, @) motion is not pseudo-elliptic ; but azimuth W and hour angle ¢ change place (0, ¢, w the Eulerian angles), and sin 6 exp (6 —A't) 1= ¥ (sec 03—cos 83 . cos @) +7,/ (cos 0;— cos 6. cos @+sec 83), CR where Darboux’s h'= a V/ (4 .sec 0;—cos @;) changes place with /, or.9' with 9 ; h=h' cos 6; and 9 now zero. But an interchange again of ¢,W will give the (wy, @) motion of a non-spinning gyroscopic wheel, or spherical | 184 Sir G. Greenhill on pendulum, with h', CR and 9’ zero, in which the axle is projected horizontally, with angular velocity ah=sin? OL, and sinks down to an angle 03; with the nadir, rising up again to the horizontal, and this makes h=n¥v (4 .sec 6;—cos 03). The motion can be shown with a plummet on a thread, say about 10 inches long, to beat as a pendulum twice a second, a double beat period of one second ; whirled round swiftly, the thread rising to the horizontal position, and sinking down again periodically. Then we find U9? = AIP? = 2gl(sec O,—cos 03), v3?= 2gl sec Os, and in the conical pendulum, at angle 03, v?= 4,2. The apsidal angle is found to be Wa har+K V(1— 2x2) > m(1—§ cos? 85) as the plummet is whirled round faster. 5. But next suppose the axle OC is held at an angle 0 with the zenith, the wheel spun with impulse CR=OC, and then released, in fig. 3. The axle will start from a cusp, at (==02, and the motion in general is not expressible in finite terms as pseudo-elliptic ; but it will represent a gravity brachistochrone on a sphere. To make the axle move steadily at the inclination @ with constant precession pw, the impulse vector CK is applied perpendicular to the axle, such that wsin@ being the com-. ponent rotation of the wheel about the axis OK’ perpen- dicular to OC, CK=Apsin @ (the inertia of the stalk being ignored), MC=Aymcos 0, MK=Ap, KM drawn vertical to meet OC in M, with the condition GK . w= gravity couple=gMh sin 0= An’ sin 0, 2 M= ass = ae : STO then the geometrical relation CM + MO=OC becomes 2 Ap cos 8+ ne =CR, the condition for steady motion. Also OM .MC=A°n? cos9=}k? cos 8, OM.MK=2 27, so that K lies on this hyperbola with asymptotes OC, OG. Pseudo-Regular Precession. 185 Or the position of M is determined by drawing QQ’ parallel to OC, to cut the circle on the diameter OC in Q, Q’, when MQ?=+k? cos 0. This is for the small value of yw in fig. 3, and the quiet precession when M is taken close to C, the other point Q’ will determine a motion where the precession pw is swift and the motion violent. Fig. 3. OC, = 20M. MK’ = 2MK., 2 \ : Spinning upright with cos@=1, w+ eh =, M,K,.0C=20M . MK, and this determines the level of AM; in fig. 5, and provides a geometrical construction for the position of C3. For OM;.M;0,;=0OA sec 6;. M;K; cos 0;=OA. AG, so that, if B is taken in OG where AB=OA, 40A .AG=OG?— BG’?=OK,?— K,B? ; M.K, . K.D.=2 OM ° MK and the circle centre K, and radius K,B will cut off a length OE on OK,=OF =20Msz, and so determine the direction of OMCs, the axle in the lowest position. When OC is horizontal, as in fig. 1, cos 9,=0, CH?= 200. CKeos 652 CK? KH KC. fae ron; one, OC — OM.- NK, — 2MK. 0,-+ 0,—= rr, the axle oscillating to an equal angle above and below the horizontal. 11. When the cusp motion is pseudo-regular and in small loops, it can be projected on the tangent cone of a sphere in a series of small hypocycloidal branches, and the motion is realised as discussed in the Principia, Book I, section X, when the tangent cone is developed into a plane and oravity radiates from a centre. With the axle horizontal, a necklace of Peeiiaroeneond ceycloids is formed round the equator, with mean regular precession yw, fluctuating in azimuth between 0 and 2uy, with azimuth interval 2¥= 2 Ke, tending as the rotation R and axial impulse CR is increased to 7 cos J,, in a.zone above the equator of angular width $7—0,, and area 27a’ cos 0, on a sphere of radius a, and the number of cycloids in the necklace would be about 2 sec 0y. 12. Even in the steadiest smoothest Regular Precession a close scrutiny will reveal under the slightest disturbance an almost invisible deviation from a perfect circular motion, in the shape of a progressive motion of an apse line, realised 190 Sir G. Greenhill on easily with the thread and plummet; utilised by Newton to illustrate the Evection of the Moon. Returning to the general unsteady motion of a top in § 8, and its vector impulse OH, the velocity of H imparted by the gravily couple An’sin@ is horizontal and along KH perpendicular to the plane GOC. This velocity is the rate of growth of KH, added to the velocity of K carried round by the plane GOC with angular velocity Be so that, with KC = Asin go and putting a = (), Cate da eee, dp -,dQ “GR IKE Ant so 0 Aaa ann nae GK = OC = 0G cosa KC 2 ees sin @ sin @ J dGK KC ak. Gk a dé sin@’ dé sin 0% obvious geometrically on fig. 3. Differentiating with respect to 0, with dd=Qdt, LdQe ae g_ K+ KO*—GEK . KC cos 0 Oat ge eee Bee) ale GC?—3GK. KC cos 0 ace A? sin? @ / 2 = G + oo 2) sé@— a ‘ an exact equation. In a state of perfect Steady Motion of Regular Precession, () and a are zero, and, in dynamical units, from § 5, OM 7» OM m2 eX NOE ah aie Natasa ang OM. ON= A?n?, A eee ON ee where A is the height of the equivalent conical pendulum, Ne= 0) ONG occ — a OG. n m OC on Oy 9. ee GO OK? sin? 8 ne (3 + cos ie, n 2 At = aa 2 +2cos0+ H )sin® C. Pseudo-Regular Precession. 191 Then in this fed Motion, 2 Q ee +m?=0, where xs OK? AP? has —4cos0= * 2 0080+ 5 = OA OP? Lame OP? m2 AP? Oe: This result is exact and reached without any approxima- tion: and the slightest disturbance will give a nutation Q=Q) cos (mt+e), beating m2 times a second, and the apsidal angle, from node to node, is on Ae | In Darboux’s representation of top-motion by a deformable articulated hyperboloid of the generating lines, the model is flattened into a rigid framework for Steady Motion ; and KM, KN produced to double length at 8,8’ will make the — focal line SS’ parallel to MN; this will be revolved about the vertical line ON with constant angular velocity uw. The small nutation will be due to a slight play or backlash in the frame. 13. The same argument can be applied to the invisible oscillation of a Simple or Spherical Pendulum, or to the apsidal angle of a particle describing a horizontal circle on a smooth surface of revolution about a vertical axis. Taken as the axis Oy, the general equations of motion of the particle are Lae) Lay L of Saat 5 Sats +5 ee +gy=H (energy), and 2? oY =K (impulse) ; so that, with 4 — - =Q, and ae ee dy?\ | 1K? =) === — : 50 ee )+5a +gy=H; and Sg, with respect to x, dx=Q dt, qt = a) BO Fee . (1+ 95) + gine d?y eae 2G da dx? at Tax Ag 0 Aes 4 9 AQ dy dy d’y Vy 3 Q dt? oe dt da dx? ( ) exact equations. —_ 192 Pseudo-Regular Precession. In a state of Steady Motion in a horizontal circle, in ao ee | Q=0, dt =(0, K=pr ’ da . d. g=wag=w NG, vt apatmgos” =9 NV, if the normal and tangent at P meet the axis Oy in G and V; and then with a =tan 0, da D) 2 : 2 ; a +m’=0, F = (84+ NGS) cos? 0; 9 oscillates between close limits a+, O= (a+) sin? $mt + (a—BP) cos? 4mt, Q=mB sin mt ; and the particle beats m/2a nutations per second, syn- chronizing with the beat of a simple pendulum of length r=g/m?, where | } Lh or ay ane 1 a Giver is a foaey fcio ty where PR is the semi-vertical chord of curvature upward of the profile curve of the surface. Then on a cone X=4GV ; and on a sphere or spherical pendulum, PR=NG, and to radius a, ~=3 cos 6+sec 8. For a profile given by y=cz”, a) dy n2, NG= Be ; v=¢g. NV our dn cae dy my dy nm—lyy , reas Ay 0G, da? i eae TNG ise Sailing n+2° Thus on the surface of a free vortex, where the angular impulse wv is constant of all annular elements of liquid of the same volume, 2’y is constant, and n=—2,rA=a0. On a motor or bicycle track of this shape, the steering will be easy, and a change of place can be made without difficulty or danger, as with the annular elements of liquids in the vortex volume. A start is made with moderate velocity from the circum- ference of the track where the slope is slight, and the car The Binding of Electrons by Atoms. 193 is steered with increasing velocity down towards the middle, where the cars can pass and repass without difficulty. To avoid a deep hole in the sink in the middle, the profile ean change to the parabola of a forced vortex, where v=pr, n=2, y= Ip NG=p, A=4GV=ISP. On a horizontal circle of this track of one lap to the mile, NP=840 feet ; described in two minutes at 30 miles an hour, NV=60 feet, and cot 0= pes = 14, a slope of 4°. Raise the speed to 60 miles an hour on this track, NP=420, NV=270, feet, and the slope is nearly 30°, the circuit of two laps to the mile made in 30 seconds. At a speed limit of 90 miles an hour, NP=280, NV =540, feet; round a circle of three laps to the mile, on a slope of over 62°. The surface could then change to a paraboloid, with a flat area in the middle, where a car could come to rest. XIII. The Binding of Electrons by Atoms. By J. W. Nicuoxson, F.2.8., Fellow of Balliol College, Oxford *. CCORDING to the quantum theory of atomic structure and of the emission of line spectra, the paths of the electron in the atom vary according to the particular co- ordinates used in the process of quantizing the separate momenta. ‘Thus in the simple case of a hydrogen atom, containing a nucleus and one electrun, we may use either spherical polar or parabolic coordinates, and the admissible orbits are entirely different in the two cases. Yet the final values of the atomic energy are the same, and consequently each method yields the same theoretical spectrum. It has been suggested that there is in fact, in every case, only one type of coordinates which can be used, when all the modi- fying circumstances, such as the variation of the mass of the electron with speed, are taken into account. ‘The only pro- blems yet solved are those in which the separation of variables, after the manner of Jacobi, can be effected, and the contention is in fact that there is, in every case, only one set of coordinates which allows this separation, when non- degenerate cases of the motion are discussed. But it is generally believed that the atomic energy is in all cases determinate and definite. We shall show, in the * Communicated by the Author. Phil. Mag. Ser. 6. Vol. 44. No. 259. July 1922. O 194 Drs: W. Nicholson on the first place, that this conclusion requires modification when the path extends to infinity. The hyperbolic orbits of Epstein, which have been used extensively in the inter- pretation of certain groups of y rays associated with many of the chemical atoms, constitute an instance, and we shall show that they rest on a mathematical-error, and that in fact it is not possible to preserve finite phase-integrals in the process of quantizing the momenta. In fact, it appears that the whole process is only applicable to finite paths, and gives no clue to the phenomena taking place during the binding of an electron which comes from a considerable distance. In another form, the question we propose is as to whether a hyperholic path is possible in the same way as an elliptic one. Such would, of course, be characterized by a posotive energy W. Certain available evidence of a simple kind, apparently not hitherto noticed, is in existence. For the existence of such paths involves the existence of parabolic paths, with W=0. In passage from a stationary state of energy W, (negative) to a parabolic path taking the electron outside the atom altogether, a quantity of energy W, should be involved. Spectral lines given by hy = W,, where W,, corresponds to any one of the stationary states, should thus exist. In other words, the ‘limits’ of spectral series should themselves be spectral lines. But there are two reasons why evidence on these lines cannot be decisive, especially when it is negative evidence. For in the first place, the values of W, determining the limits of series are of such magnitude that only for two or three, in any case, can the corresponding lines come into the visible spectrum, and with only hydrogen atoms and charged helium atoms to test, and enormous band spectra for both elements, the test cannot readily be applied. Moreover, the probability of _an electron entering the atom in a parabolic rather than a hyperbolic path is so small that any resulting lines could hardly be expected to be of visible intensity under ordinary conditions. We consider, therefore, that the question whether limits of series are themselves spectral lines, on the principles of the quantum theory, cannot, at least at this juncture, be examined in the light of experiment, and that it must remain a matter of deduction from other phenomena. We find it necessary, as stated, to disagree with the hypothesis, explicitly indicated several times by Sommerfeld and others, and implicitly assumed at least by the remaining Binding of Electrons by Atoms. A195 writers on the quantum theory of spectra, that the energy W is always completely determinate when all the momenta are quantized. This can be disproved not only for fictitious laws of force in an atom, but for laws which must actually occur in systems with an existence, if only a temporary one. Consider, for example, a simple doublet and an electron in orbital motion about it. Regarding the doubiet as stationary, and of moment M, its external potential is when it is situated at the origin, with its axis along the axis of <, using spherical polar coordinates. The equation of energy for an electron moving in its presence is Me cos 0 lm {i+ 16? +r? sin? ee a —W. The momenta are, in the usual notation ») >] fo hh . EU yp cee, sce ON Dliaig: SUNT: Da Vial mr, or ot Pee, ee | mr sin Od, He ae, b so that um | pe sp Be igo ae eee —W. eS sin Now ¢ is a speed coordinate as usual, so that Ps = const. = mh/27 when subjected to the quantum relation, n, being an integer. For the Jacobi solution, we must also take, in separating variables, 2 te A + 2mMecos? =8 where £ is constant, and 5m | pt Et =— n=a/—Imw— 8, Thus QO 2 196 Dr. J. W. Nicholson on the With a positive W, the motion is not real. Thus W must be negative and the path necessarily extends to infinity. A critical value of 7 is \/ ee and the other is infinity. The phase-integral for p, 1s 2 pdr B —2mW which is infinite, but nevertheless independent of W. For writing — ts i =;.6, it becomes —2mW sts a) He / = ¢ Ve aes ti 2 { V —2mW (1 =) ~amw 0% == 2 vp{ da(1— 2) A finite integral is secured,—Hpstein’s procedure, :for instance,—by, using the phase-integral not for p,, but for Pi—(P1)r=0, Which in the same way yields 2 va ae f(i-4) 1} = 2v7 (1-5): again independent of W. Now 8 is quantized, or expressed definitely in terms of integers already, from the phase- integral for the momentum p,. The phase integral for py can only, in this case, lead to another expression of similar type for 8, but to no expression for W. Itis not at all clear that the two expressions for 8, also, can both be valid simultaneously. This possibility has hitherto apparently been overlooked by authors in this subject. No case has, however, been noticed in which W is inde- terminate for a finite path. One very important conclusion is that the whole investigation is valid for a negatively charged atom with a distant electron. We proceed now to discuss the possible existence of definite paths with a positive total energy and infinite extent, for a single electron around a nucleus of charge ve, situated at the origin. This is Epstein’s problem, which he treats as only two-dimensional. The energy equation is 7 Lr Binding of [lectrons by Atoms. 197 where W is positive, and represents the total energy, and the p’s are the momenta. We have thus h p= const.= 2a? ate Da = 9? P2 sin? 6 ; being clearly positive, 2 2 4m {pit = } ps ee ; , The phase-integral for py, is aA pedo dé ‘des aE all Pa y p sin? 0” the limits being the suitable values of @ for which p.=0. The factor 2 represents the double journey in this co- ordinate, sin = 3 where wy is one of the limits, and the other admissible value, for a real integral, is 7—y. Thus mY Rah = 2 { dé / 8’ —p7/sin? 6 B — ap ( wl { with sin 6 =3 vo). 1 fox w = sin? w+ —5C0s’ a, P3 and we have fs —n3" 1/2 nh = 48 -+——— YE Dana 4 cos? w dw B? ’ sin? wm + —; cos? wo 2, . P or with tanw =f, 4 3 Rey at ngh = 75 (8?—p,?) (| ——4, — P3 J0 a+e)(", +2) = 48 { tan i— Pian} 0 = 2084 1-72 | = 2n(@—ps), 198 | Dr. J. W. Nicholson on the whence h ] C= se a P23. (mitms) 5 these integers being thus additive, in the usual way. The phase-integral for jp, is : i 2m ve? 2 3 —— dae 2m W + ie 3 7 72? if we seek to quantize p, as it stands. The limits would then be a positive value of r and infinity, for half the path, and the integral would be infinite. But it is clearly necessary to suppose that when the electron is at infinity, out of range of action of the nucleus, it should not be subject to a quantum relation, so that (p;)-=« 1s not affected by the rule, and only the variable part Pie Pie is so affected. Yet this question of quantizing p, presents some difficulties in whatever way it is suggcsted that it should be effected, and we consider that Hpstein’s discussion of the matter is very incomplete and not logically justifiable in its mathematical procedure. We shall thus consider various alternatives which may give a finite phase-integral. Now the actual r-path is not a passage from r=« (say) to y=oo and back, and the phase-integral is not twice the defiuite integral between these limits. The electron goes from a limiting radius to infinity, and back to the same radius elsewhere, and the passage through infinity distin- guishes this phase-integral from those which occur in the other coordinates. We must, of course, also remember that the sign of p, depends upon the part of the path concerned ,--whether the electron is departing or returning. The critical value of r is the positive root of Pave ea p< 2mW + —— , P 1 _ mye? + /mv?e4+ 2mW PB? 1 ig ig PCE Lee NGA MITE AC ie re (say). or Writing, a an with a new variable 4, _ me? mye! + 2m WB? + oe = cos qh, Binding of Electrons by Atoms. ivd we have @=0 in the critical position (perihelion, in the usual terminology), and mve eee nW B? sega ETEONB uy ay) when r= What is required for the correct evaluation of the phase- integral is a continuous variable which shall change in one direction,—and thus give a definite integral,—as r goes to infinity and returns, the sign of p; being automatically taken into account,—or the sign of py;—(pi), when (p),, is not zero as in a parabolic path. The new variable ¢ has this property, and ranges from zero to 2m as r goes through its changes. We have denoted its value, when r=m, by 7 above, where 7 is evidently an obtuse angle. The phase-integral for p; alone would be PE evils Wee ery De ngh =| a dr x / 2mW + —— TE — (the square root being properly interpreted in different regions) where 1 = mve? e 8 V/ mpv*et + 2mW 8? cos d 0= ee + = V mvet + 2m WB? COS 0, and we find ‘ 5 AMUN ee, dr= — uy ae » g= VIMW 4+ me, (cos@— cos)” yg 2 a/2nW + . — = = gain dp. If the integration were continuous throughout,—as as- sumed by Epstein, —we should thus have hae ae meen ap (cos @— cos 7)? =26(" _ sivtddd _ (cos @— cos 9)?’ which is an infinite integral, as would be expected. 200 Dr. J. W Nicholson on the If we merely quantized over the finite part of the hyper- bola,—another possible suggestion,—we should have oe sin? ng 8 he - (a (cos @—cos 7)? a 7 sin? ddd 9 (cos d— cos 7)?’ which is again infinite. The nature of the first infinity merits a remark, however, for it is independent of » and therefore of W. For 72) ie sim oddb __ sin a ee cosh db Jo (cosd— cosy)? Paee ah 9 CosSp— cosy = [ an | — 7 — COS af” ode cos d6— cos cos }— cos n- —= 7) em The principal value of the last integral is well known to be zero, for all values of 7, so that the last term is zero. Our equation would be nh= —278 + | any cos d— cos where the principal value of the bracket must be taken, 2. €. it is to be interpreted as fee This becomes sin sin) Le a ina Se oe ir Wen which, though infinite, is an infinity independent of 7 and therefore of W. We have another aspect of the indeter- minateness of W for such paths. Our fundamental objection to Epstein’s mode of integration may now be introduced. He integrates p,;—(p;),,, and not p1, but this fact does not affect the question. Foras ¢ ranges between 0 and 2m, if p=/($), we have p, varylng con- tinuously with 6, and remaining positive, till@=7. Then p, becomes —/( (dy) when ¢=27—d, on the return Journey after 6=2r—n. Between d= and d=27—7, the value of r should be infinite, and p, changes from 2mW_ to — /2mW, as in the figure. Binding of Electrons by Atoms. 201 The variation of p, between + “2mW at infinity is the source of trouble, and it takes place while : =A) O= 27-7 Epstein takes twice the integral from ¢=0 to d=z7, but according to the substitution formula, r is negative when $ goes from 7 to 7, and negative values of r are clearly not permissible. A suitable integration for the infinite region cannot in fact be effected, and any supposition of a suitable variable in place of ¢, for the change of p, at oo from /2mW to — V¥2mW, would be entirely arbitrary,—but as it could not lead to a finite phase-integral, we pursue the matter no further. ‘These considerations, nevertheless, have considerable force when, thrown back as we now are upon the necessity, if the quantum theory is applicable, of using p;—(p,),,, we attempt to quantize this. We have, when 0=n (pi). = ¥2mW — 2 sin N; 8 where g= Vmv?e' + 2mWB? as before. And when ¢=27— 7, ® (pr). = sin (24 —m) = — 4 sin 7. From ¢=0 to d¢=7, Pi (Pr) = B (sin $— sin). From ¢=27—7 to $6=2r7, Pi- (Pi). = + A sin p+ sin), and from ¢=7 to $6=27—7, Pin Pu. O- 2020 Dr. J. W. Nicholson on the With the value of dr, the phase-integral 2a ngi=( dr( py — (P1).) breaks into three parts, thus nh =p)" sin ¢ (sin @— sin 7 ae 0.d¢ _(cos @— cos 7)” ue sin @ (sing + sinn\dd +0(." He (cos @ — cos 7)’ ee S07) a 5 og\ 0. dd (cos @ ~ cos)’ by a simple transformation. Finally. the only accurate phase-integral is nah ee ve sin @ (sin @— sin ”) 4 ae, (cos 6— cos)” while Epstein gives, iu our notation, | a “® sin d (sin d— sin 7) nah = 26) (cos d— cos)? dp, the part of his range from to 7 involving a meaningless negative value of r, and violating p;=(p,),, though the moving electron is at infinity. The principal value of Hpstein’s integral is, using the indefinite integral for the function in the form, readily obtained by parts, {= @ (sin @— sin Le (cos d— cos n)? _ sing@— sin ~ cos @— cosy Je a cotn. ac sin; of the type 9 ° nzh=28q { ols ee yy (ny + No a 13) or 7o aoa wT T sin 9 Ny + No and ultimately Qian ody if we sag fae q (mine +3)? — (my +79)? —generalized from his value which relates only to a plane Binding of Electrons by Atoms. 203 hyperbola. We have the sum 2+, of the angular quanta in place of his single integer. But this formula, with all the applications he makes to characteristic y radiation, is not tenable, as resting ona mathematical error. Its apparent success appeared at one time to the writer to justify it as an empirical formula, in spite of his independent investigation, outlined above, indi- cating the impossibility of quantizing such orbits. Close examination, however, of the calculations of y radiation and so forth made it clear that they were in several cases illusory, and determined more by order of magnitude than by the nature of the formula. There is one convincing argument against the formula, however. It should give an emission spectrum for all values of ny, m2, nz and m4, mo, m3 making © — W(m, mo, m3) + W124, 2, 73) positive. This can be tested in great numerical detail on the spectrum of a hydrogen atom, and the test fails entirely. No spectrum line is found,—in the secondary hydrogen spectrum,—in any of the assigned positions. Thus the formula really fails as an empirical one. We have seen above that it must be replaced by nah = 28 ik sin (sin f— sin 9) dd 0 (cos ¢ — cos 7)’ sin gd— sin | 2 = 28 | —",-——_ — $+ cotnlog,< — cos @— cos n ane 2 0 which is logarithmically infinite. The attempt to obtain a finite phase-integral, in this manner, in fact fails, and we must give up the hypothesis that even the variable part of p, can be quantized for the infinite path. It is not difficult to see that this conclusion is general for any infinite path which is possible for an electron about a physically existent atom, whose nucleus can always be regarded, for the present purpose, as a superposition of free charges and a set of doublets. We have demonstrated the result for a single free charge, and previously for sets of doublets. Further analysis of the more general case does not seem necessary, and could readily be supplied by the reader. Our conclusion must be as follows :— A determinate and finite value of W cannot be obtained for an electron moving about any atomic nucleus, if the path involved takes the electron to infinity. Lio e.] XIV. Theoretical Aspects of the Neon Spectrum. By Laurence St. C. BRouGHatt * \HE object of this paper is to attempt to explain the - spectrum of neon in a manner somewhat similar to that used by Bohr f in his explanation of the reason for the existence of the Balmer series in the hydrogen spectrum. The principle on which this hypothesis rests is that when an electron rotates in a fixed orbit it does not radiate energy, although the principles of electrodynamics state that it should; if, however, the electron changes from one orbit to another, then energy is emitted, provided that the kinetic energy of ‘the electron is less in the second orbit than in the first. In order to account for the spectrum, it is assumed that the energy emitted is numerically equal to the product of the frequency of the spectral line produced and the quantum constant. We thus obtain the equation K=nh, where Hi = energy emitted, n = frequency of the resulting radiation, and 4 = quantum constant. c In the case of hydrogen, it was assumed that the orbit of the electron was circular, and then the attractive force between nucleus and the electron due to their equal and opposite charges was balanced by the centrifugal force of the electron due to its rotation about an axis passing through the nucleus. The energy of the electron can thus be found for any radius of orbit. When the electron changes its orbit, it moves to one with a radius which is an exact multiple of the radius of the original orbit. In this manner the change of energy due to a change of orbit can be found, and then, using the equation given above, it was shown by Bohr how the constant of the Balmer series could be found; and the value so obtained agreed extremely well with that found by experiment. In the case of neon, we are dealing with an atom which contains more than one electron; and since the atomic number is 10, it follows that if the atom is to be neutral, then there must be 10 electrons present to annul the excess. of 10 positive charges in the nucleus. * Communicated by the Author, tT Phil. Mag. vol. xxix, p. 382. Theoretical Aspects of the Neon Spectrum. 205 In order, therefore, to study the atom, it is essential that the electrons be given definite position relative to one another. This has been undertaken by Langmuir”, and there is considerable evidence in favour of the postulate that eight of the electrons arrange themselves at the corners of a cube at the centre of which the nucleus is situated. The other two electrons are imagined to lie within this cube, probably on a line joining the mid-points of any pair of opposite sides. If we make use of this hypothesis, and further are in possession of data which will allow us to find the length of the diagonal of the electron cube, then it was shown by the author{ that it is possible to calculate the angular velocities of the electrons about the nucleus. Since the determination of the spectral lines is an ex- tension of the matter given in that paper, it will be advisable here to state the principles on which the calculations of the electron frequencies depend. It has already been stated that the two inner electrons will probably lie on a line joining the mid-points of any pair of opposite sides. If this be the case, then the electrical forces acting on the outer electrons due to the other electrons in the outer shell, and to the two inner electrons, will be the same whichever electron we take, provided that the two inner electrons are equidistant from and on opposite sides of the nucleus. The next consideration was the axes of revolution. As before, it was desired if possible to get the forces acting on the outer electrons due to centrifugal action the same for all of the electrons. If we take as axes the three lines which pass through the mid-points of the three opposite pairs of sides of the electron cube respectively, then the above con- dition will be satisfied. In the diagram the axes of revolution are illustrated by XX’, YY’, and ZZ’. The inner electrons being on the axis XX! will only rotate about two axes. It is, of course, quite immaterial which axis the inner electrons lie upon. The forces acting on any outer electron were then considered, and were taken along the three sides of the cube which meet at the point where the electron is situated. Now, since the electron must be in equilibrium, so the force along each of these lines due to electrical attraction and repulsion and also due to the motion in a circular orbit must be equal to zero, * General Electric Review, 1919. *, Phil. Mag. Feb. 1922, 206 Mr. L. St. C. Broughall on Theoretical In this manner three equations were obtained, namely : gel) 1a ene e*(r +l) e(r—lL) 4s — 4 t 98 + [n+l +s2]22 ~ [(r—1)? + s?}9? +mo t+ mel, . . db) 3981 oP el Le Ae aap. woe 6 (o+ lets 3? tT (leer : me 71+ mes le.) nll) DOE 7.) e7U e7] e7| Wet tos + (eae (Gee yas + mast, s 2 eaten Another equation can be obtained by considering the forces acting on either of the inner electrons along the line joining the two inner electrons. Equating the forces to zero, we found that Mr ee ne) een ee e — fer)? +s 22 t free? t ae +mor+mo’r, . . (LV.) In the above equations, e= charge on an electron ; Aspects of the Neon Spectrum. 207 ry = radius of orbit described by the two inner electrons ; @, =angular velocity about YY’; w,=angular velocity about ZZ'; w; = angular velocity about XX'. |= 43 length side of electron cube; s = $ surface diagonal of the electron cube ; c = 4 diagonal of cube. ‘s’ and ‘l’ are of course functions of ‘c, and if the latter is known, then ‘s’ and ‘1? can be found. m = mass of an electron when its velocity is small compared with that of light. - From these equations it was shown by the author that an equation involving only ‘7, ‘ce,’ ‘1, and ‘s’ could be found. In the paper mentioned, ‘/’ and ‘s’ were not expressed as functions of ‘c,’ but expressing them as such, since /= nee and, s= ee. we obtain the equation oe r+°d7Tc f;_ 2} pom Neco ie)?-- *66fe? |" r r—‘d77Te f 2°308¢ D°63¢ ~ [ (r—-577c)? + -667c? |*? vai r } caer ele (V.) In a recent article it was shown by Prof. W. L. Bragg * that the diameter of the neon atom could be found by an inspection of the diameters of the atoms of elements whose atomic numbers were near that of neon. It is impossible to measure the radius of the neon atom directly, since it forms no chemical compounds. ‘The value obtained was very much smaller than that found by gas measurements, and the former is considered by Bragg to be the distance between the elec- trons in the atom—that is to say, is equal to ‘ 2c.” The value obtained by Chapman 7 from gas measurements is, however, the diameter presented by the molecule when in collision with other molecules. ‘The difference is due to the fact that in molecular collisions in the gaseous state the outer electrons of the molecules do not come into contact tf. Using Bragg’s value we have 2c=1°30 x 107° cm., and on substituting in equation (V.) we have a means of obtaining the fundamental value of ‘7.’ An inspection of equations (II.) and (III.) shows at once that mj=@, and equations may be obtained for @, and a3. * Phil. Mag. vol. xl. p. 169. ft Trans. Roy. Soc. A. vol. 216, p 279. t Rankine, Proc. Roy. Soc. vol. xeviil. p. 360. 208 Mr. L. St. C. Broughall on Theoretical These aa take the form es rt r—l Imr {i a ktces) a SLB T Fp S(r =e] t = 04" ; (VI.) er ih it a. hese 1G Hae =,'—o;. (VIT.) Using a slightly different value for ‘m’ from that used in ~ the previous paper, we obtain the following values :— i, w,(=,). ol9x 10-° em. 6°034 x 10 radj/sec.. 4290 x 10™ madi zee ‘m’ being equal to 9°005 x 107-3 orm. and e=4°774 x 107 ES. These figures refer to the neon atom when in its normal state. There is some donbt as to whether they apply without modification in the gaseous state, but certain assumptions are made later in this paper which leads one to the conclusion that if the atom is larger under natural conditions, then the only result will be the elimination of certain spectral lines in the ultra-violet. When the atoms of the neighbouring elements were submitted to measurement, they constituted a solid body ; it is, therefore, quite conceivable that modi- fication will occur if the element becomes gaseous. In order to explain the nature of the spectral lines, we have to consider the change of energy due to a change of orbit, energy being emitted when the orbit increases in diameter. Bohr, as already stated, imagined in the case of hydrogen that the radius of the orbit increased by constant multiples of the radius of the initial orbit. To adopt such a plan in the case of neon would lead to the emission of spectral lines of a frequency which would only give ultra-violet lines under reasonable circumstances. Further, there is no reason why the increment should be of such a nature, and the hypothesis used in our case is that the spherical shell formed by the inner electrons increases in radius until the shell has a radius equal to that of the initial outer shell of electrons. In order that equilibrium may remain, it is essential that the outer shell also expands to an extent which can be calculated from equation (V.). | The initial increment is of the nature of 3x1071 cm. This process of expansion continues again and again, the inner electrons always occupying the orbit previously occupied by the outer electrons. Aspects of the Neon Spectrum, 209 Having thus found a definite value for ‘r’ corresponding to a detinite value for ‘c,’? we are now in a position to calculate the values of the angular velocities about the several axes. To do this, equation (VI.) is first applied ; and having found ‘@,’, the value obtained is substituted in equation (VII.), and @;? obtained. In calculating the value of ‘¢c,’ use is made of the fact that the ratio ¢: 7 is practically constant; such ratio values are used in Table II. The closeness of the ratio figures and the absolute figures is shown in the appended table. TABLE I, r. ¢ (ratio method). e (from eqn. V.). 650x10~° cm. 683X107? cm. 683x107? cm 1233). 1295. 1295, yh) eae 1577) 045 Lite ols It wili be seen that the agreement of the figures is so nearly exact as to warrant their use, remembering that our fundamental value of ‘c’ has not been obtained experi- mentally. Table II. shows in columns II. and III. the values - c? and ‘r,’ and in columns IV. and V. the values of w,’ and w;” respectively. c TABLE II. | | I. | II. III. IV. V. | Radius Radius 5 2 | No. | Outer Orbit = c. | Inner Orbit =r. na ty it | 1... 650x10-%cm.| 619x10-%cm. | 36-41 1082 | 18-40 x 10°? | SS Sam ae) 3141 ,, | 15°84 apt Was (oe 68s) 4); TOS A WISER eget Ve es Tiger, 2 tae a ey f= a ay gs ate foi cae BAG | TOR os A? eee 7 Vesicle Ses”, ONG? >. Cie aos 1500 ,, 7-568, 8 = is. 3. S74, 19-05 6528, x 964; i ae tics. E20", . | es Soe ee 964 9625 ;, | 4860 ,, | H1,..| 1084" IGS! |, 8-299 ,, 4-182 ,, Gi a aes i 7-160 ,, 3613. ,, aa a er iris’ |, 6170 ,, 3109 ,, | A 18S oy £YFAO iy 5326 ,, 2684 ,, | Bees 1295.1, 12:43...) sy, 4°602 ,, 2-322, 1s en” | 12:95, ty ae F002, | yin i > 1360 _,, 3:428 ,, 1s et! a 163) GeO ©, 1429, 9-954 ,, 1489 ,, | ee G7 ae 1501, 2°550 ,, 1-286. ,, aS > saree” 5, ey, 2-201 ,, 1113 21 | Cine Oe 1657, 1897 ,, 9581 ,, Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. P 210 Mr. L. St. C. Broughall on Theoretical Let us now consider the energy of an electron in the outer shell when the diameter of the shell =2¢. Let the angular velocities about the axes YY’ and ZZ’ = VW. Let the angular velocity about axis XX’= Ws. Since the diameter of the shell is equal to ‘ 2c¢,,’ it follows that the radius of the electron orbit =8, where S;=¢, V/ 6/2. Using the above notation and remembering further that the kinetic energy of a particle describing a circular path of radius ‘R’ with an angular velocity ‘W’ is equal to 4MR?W?, where M is the mass of the particle, we find that the kinetic energy due to rotation about the axis XX’ =imS,?W,’, and the kinetic energy due to rotation about the axis YY’ plus that due to rotation about ZZ’ | =mS,;7W,? since W.=W2. Therefore the total kinetic energy of the particle is equal to 3mS77f2W,?+ Wt. In the case of an electron in the inner shell where the radius of the orbit =R,, we have the kinetic energy of the electron due to its rotation about the axes YY’ and ZZ’ =K, where Ky = mR,’ W,?. Now let the inner shell expand until it occupies the space previously occupied by the outer shell—that is to say, until R,=C), then kinetic energy in new orbit = Ey, Where He=mO/?W,?, ‘Wy,’ being the new angular velocity about the axes YY’ and ZZ’, the change of energy, Tm Ky, ne K,, = m(RYW — CW") ° Meanwhile the outer electrons have moved further away from the nucleus, and now the outer shell has a radius =C,, and the orbit of the eleetrons is now S,. 3 Therefore the energy in the new position is equal to im,” [2W4,? + W37 | ° Where ‘W,,’ is the new angular velocity about the axis dlspects of the Neon Spectrum. 211 XX', the change of energy due to change of orbit is therefore equal to tm{S?[ 2 WY + V ical — S.?[ 2W,? + W37 | Pi: Now, by Bohr’s assumptions we have the equation Energy Hmitted = Frequency x h. Now, the frequency of a light-wave =c/X where ‘¢’ is the velocity of light and ‘2? is ‘the wav e-length. Therefore Hinergy Emitted = e= a or —— ch € Now, in the case of an inner electron eE= m(R,? WY — C?W;,,”) 3 Ss ch oe m(h,? W As ae Uy? W 11’) : giving a series of lines for different values of R. In the case of the outer electrons, 2ch giving a second series of spectral lines. Table ILI. shows the energies corresponding to definite radii. Column II. shows ie energy content of an inner electron on the left, and that of an outer electron on the right. Column III. shows the change of energy, and column 1V. shows the wave-lengths of the spectral lines produced. It should be stated here that the energy under consideration is the energy of one electron and not of the whole shell. It has been stated that there is a possibility of the atom not being in its normal condition to begin with, owing to its gaseous condition. If, however, it has expanded, then instead of starting with an atom whose diameter is 1°30x10-8 cm., we start with one whose radius is in all probability equal to one of the radi given in Table IJ. If this is so, then the only change produced will con- sist of the elimination of some of the lines of higher frequency. P2 212 Mr. L. St. C. Broughall on Theoretical Taste III. I Ila. IL 3, IIL a. TIL 2, | Wao aes No. Energy per electron. ace ate nee oe Inner. | Outer. Inner, Outer. Inner. | Outer. 12255 X10 1-157 x 10-0 | 2.11195 , |1101 , |80X1077* 56x10~™) 3272 A° 3505 A® g..j1137,, [1048 ,, (58 » [58 4» | 3886, | 8704 ,, ie eo) 970 ,, |25 » [51 | 3570 ,, | 3850 ,, cree 49], |52 » [479 4, | 8776 ,, | 4100 ,, 6 | 9801 9035. [496 » (456 ., | 3959 ,, | 4305 ,, Mr Ve iggay 604 ,, [£67 » |481°,, | 4205 ,, | 4555 ,, 8 ...| -8886 gis7 ,, [#51 » {417 ., | 4853 ,, | 4709 ,, Oe einy 7793. [#29 3, [394 4, | 4576 ,, | 4983, LON "8055 __,, 7423 fs 4-02 Ai 3°70 35 4884 5 5307 y igo) -veGo. ‘7063, DOOR AE BGO" Se 50384 ,, | 5454 | 12 | -7297 , | e723, [868 » [840 ., | 5885 ,, |-5775 ,. 13 .../ 6940, | 6396 ,, [857 » (327 | 5499 ,, | 6005 ., 14 ...| 6609 ,, | -eos9 ,, |88l » [307 ., | 5982.,, | 6396 ., ib ..| 6298 ., | 5903, Jetl » |286 ., | Sale), ieee 16 ...| 5995 ., | -55g5 , [3038 ». {278 .. | 6480 ,, | 7063 ,, 17 ...| -5709 ... -|--59g0. «|288 » =) 285s, 9) 6864 | Oa Jig ...| 5430 ,, | 3003 ., |279 » |257 » | 7037 ,, | 76205, 19 ...| 5172 ,, | 4766 ,, [258 » {287 ., | 7610 ,, | 8285 ,, 20 ....| 4997, | 4540. [245 » [226 », | 8013, | 8688 ,, 91 ...| *4689 nf 439] vi 2°38 25 2°19 a9 §249 a 8566 Bout Owing to the complexity of the neon spectrum, it would be useless to attempt to compare our calculated lines with those found by experiment ; indeed, it would be deleterious to attempt such a comparison, since the impression would be given that there is a definite line in the spectrum which corresponds to one of our calculated lines. Hmphasis may only be laid upon the fact that our series produce lines in the visible part of the spectrum, which do not compare unfavourably with those obtained by experiment. Reference to Table III. will show at once that only forty lines have been determined between 7X=3272 A and 7A=8966 A, Aspects of the Neon Spectrum. rp whereas there are many more lines in existence. These lines can only be explained by the fact that when the spectrum is obtained, large numbers of ionized atoms exist, and under such circumstances our fundamental equations no longer hold. II. Jonization in the Neon Atom. The discussion of the properties of ionized atoms is very complex, since the possible degrees and modes of ionization are very numerous. The first case which comes under con- sideration is the atom which has lost one electron, thus leaving an excess of one positive charge. It is very probable that one of the outer electrons will be removed, thus leaving seven electrons in the outer shell. Now, it seems probable that the angular momentums of the remaining electrons will suffer no change, the light pro- duced during ionization being due solely to the change of energy of the electron suffering removal. We are thus left with seven electrons, each possessing the same angular velocity. It is a matter of considerable difficulty to arrange these electrons, and it is impossible to arrange them ona spherical surface without the force acting on an electron varying with the electron taken. We are therefore obliged to separate them into different shells. Now, since the angular momentums of our seven outer electrons are the same, it follows that, if they are not on the the same spherical surface, then they must be in motion relative to one another. Under such conditions the positions of the electrons will vary with time. Owing to the complexity of such a case, it seems impossible to treat the case mathematically without more experimental evidence. There are further atoms present which have been ionized to a greater extent, thus losing several electrons. Similar difficulties are met with in the cases of atoms with five or six electrons in the outer shell as in the case of seven electrons. The cases in which four or six electrons have been removed are, however, considerably simpler, since the electrons may then be given positions on a spherical surface such that the force acting on an electron is not dependent on the electron taken. In general, the atoms ionized to so great an extent will be comparatively few. In all our cases of ionized atoms, it must be remembered that it is not only the normal atom that is ionized; an atom may have given out several spectral lines before it becomes ionized. Soa 214 Theoretical Aspects of the Neon Spectrum. large number of lines will be obtained depending upon the state of the atom when one or more electrons are removed from it. There is another form of ionization which is worthy of consideration. That is the case in which an electron has succeeded in penetrating the atom and reached the nucleus, thus temporarily reducing the positive charge and therefore giving a negative ion. The fundamental mathematical expressions for such a case are found by extending our formule for the oe atom for the case where the char ge on the nucleus is ‘n’ instead of ten. | The angular velocities about the several axes will remain unchanged ; and so only two equations will be required to determine the new values assumed by ‘c’ and ‘7,’ the radii of the outer and inner shells respectively. We have three available equations ; and expressing them in the notation previously used, we obtain : (4n—1)e"l _ eee eu e(r-+1) e*(r —l) 4e8 A]? ails eo ob) (r+)? +e] ap i (r— 1)? + 922? + 2mo?l: oc 2 el Ciel eo el e7l e7| 4 AP T28T (elPee pet (Gb? +8? % +mol+mo,l, . . (Ila.) ney Ioan |) Ae? (r—l) ip P= Toth +e 2 ct (oD? +9? 1 Ape + Qo? 2.4) 3) ele which are obtained from equations (I.), (II.), and (IV.), replacing the nucleic charge of ‘10e’ by ‘ne,’ and remem- bering that w;=@». The result of the alteration will he that the electrons will move further out from the nucleus, since ‘n’ is of necessity less than ten. In consequence of this, the frequency of the spectral lines produced by such ionized atoms will be of a lower frequency than those produced by the neutral atom. There will in consequence be a larger number of lines in the part of the spectrum of greater wave-length. It is for this reason that there are so many lines in the orange, and red in the case of neon. Feb. 18, 1922. C’ \ Eaviiaes XV. Absorption of Hydrogen by Elements in the Electric Discharge Tube. By F. H. Newman, Ph.D., F. Inst..P., A.R.C.S., Head of the Physics Department, University College, Eweter *. 1. Introduction. ‘ia phenomenon of the disappearance of gas in the electric discharge-tube, and in the presence of incan- descent filaments, has received much attention recently owing to its importance in technical applications. Langmuir + has shown that hydrogen disappears from a vacuum tube in which a tungsten filament is heated above 1000° C., This fact has been utilized by him in the removal of the last traces of gas in valves,and the effect has been termed a “ cleaning up’ one. The pressures at which he worked were very low ; for example, he found that the pressure in a tube was lowered to 0°00002 mm. of Hg. Other gases, including nitrogen and carbon monoxide, are removed in a similar manner, and molybdenum, when incandescent, has the same effect as tungsten. In all cases Langmuir found that the cooling of part of the apparatus by means of an enclosure at liquid- air temperature greatly accelerated the rate of disappearance of the gases. In addition he noted an electro-chemical “clean up,” which occurred at much lower temperatures of the filament, when potentials of over 40 volts were used in a way that caused a perceptible discharge through the gas. More recently Campbell, conducting work for the General Electric Company f{ and using incandescent filament cathodes in electric discharge-tubes, has made an exhaustive study of the ‘clean up” effect, and has come to the conclusion that there is much evidence for believing there exists an electrical action which is quite independent of the thermal action, and, providing the temperature of the filament is kept below that at which the chemical ‘‘ clean up” occurs, the effect appears to be one dependent only on the electrical discharge. Inthe case of the disappearance of carbon monoxide there is proof of the conversion of this gas into carbon dioxide, and the action takes place more rapidly when part of the apparatus is cooled to liquid-air temperature. This has the effect of removing the carbon dioxide by condensation as quickly * Communicated by the Author. + Am. Chem. Soc. Journ. vol. xxxvii. (1915). t Phil. Mag. vol. xl. (1920); vol. xli. (1921) & vol. xlii, (1921). 216 Dr. F. H. Newman on Absorption of Hydrogen as it is formed. The presence of phosphorus vapout accelerates the rate of disappearance of all gases except the inert ones, and much lower final pressures are attained. This, the author believes, is due to the deposition of the gas on the walls of the vessel, this deposit then being covered with a layer of red phosphorus formed by the electric discharge passing through the phosphorus vapour. The covering of red phosphorus prevents liberation of the hydrogen by bombardment of the ions, and at the same time provides a new surface on which further gas can be deposited. The problem of the disappearance of the gas is a very complicated one, owing to the many factors to be considered. The walls of the vessel and the electrodes will certainly receive some of the gas, although the latter may not disappear in its original state. There will be chemical changes occurring in the volume of the gas, such as the conversion of carbon monoxide into carbon dioxide, and, in addition, any other elements present in the discharge-tube, either in the form of vapour or solid, will affect materially the rate of dis- appearance of the gas and the final pressure reached. The author * has shown previously that various substances present on the electrodes of a discharge-tube alter considerably the amount of gas that can be caused to disappear when an electric discharge is passing. In particular, phosphorus, sulphur, and iodine cause both hydrogen and nitrogen to be absorbed ata very great rate, and a high vacuum is quickly produced as a result. This action of phosphorus has been used for many years to obtain and maintain very low pressuresin valves. These three elements stand out as being very effective even at high pressures, but other substances which were tested in a similar manner did not appear to absorb hydrogen. On the contrary, gas appeared to be liberated. This effect can be explained as follows. At pressures above 1 mm. of Hg.a certain amount of the gas in a discharge-tube becomes occluded within the walls. This gas will be liberated when the walls are bombarded by the ions produced by an electric discharge. This effect will mask any disappearance. If, however, the tube is heated almost to the softening point of glass and highly exhausted, then on admitting hydrogen at a small pressure such as 0°l mm. of Hg., very little occlusion of the gas within the walls will take place, and on passing the electric * Newman, Proc. Roy. Soc., A. vol. xe. (1914); Proc. Phys. Soc. vol, xxxii. (1920) & vol. xxxiii. (1921). by Elements in the Electric Discharge- Tube. 217 discharge practically no hydrogen will be liberated from the walls by bombardment with the ions. If there is any absorption of the gas, this effect will not be masked by the liberation of the gas from the walls or electrodes. The object of the present work was the study of the behaviour of hydrogen in the presence of various elements in a discharge-tube when a current was passing through it. The pressures of the gas in these experiments were much lower than those used by the author in the papers quoted above, but they were much greater than those used by the previous investigators—Langmuir and the General Electric Campany. 2. Description of Apparatus. At gas-pressures below 0°1 mm. Hg. it is difficult to obtain a current through a discharge-tube unless very high potentials are used. ) ; for rigorous and thorough treatment of the latter. 66 III. CompLex SOLVENTs. The importance of this question of ‘ complex” (polymer- ized and mixed*) solvents is sufficiently obvious when it is considered that of this type are most of our best solvents, probably all our “ ionizing” solvents, and, chief of all, water. We have to consider a solution consisting of the solute- molecular species, ™m ,, mo, .....- ite... slim adaiions to. the Varlous species 791, Mp, .-.., Which constitute the solvent. Concentrations being expressed in gram-molecules per gram of solvent, the ‘“‘ molecular” expression for y is Jr bh = Sng ($u Rlog re = +3n.(g,—R lege es) BT Ye (ares 4), (GIL which is of the form Se OY an OY Vv 7 >No) N01 eed a a somes 2 (2) the several solvent-molecular species appearing as separate * Mived solvents, while submitting to the same theoretical treatment as the merely polymerized solvents, present certain peculiar difficulties and some interesting possibilities with which it is hoped to deal at length in a later paper. + A suffix outside a bracket is used to indicate, in less obvious cases, independent variables which are held constant in a partial differentiation, —a well-known usage. A single suffix may be used briefly for a whole series, aS 7; here standing for m1, %o2,.... - 234 Mr, Bernard Cavanagh on components, whereas the “ experimental *”’ expression must have the form pela oy wr = My OM, +n, au. ii ulate ee Ue eae (3) the solvent appearing as one component only. Clearly Bh Gen BI Ons, T NORe Moi ON: — Osi My but the relation we have to use in making the change of variables is that given by the condition for chemical equi- librium among the molecules of the solvent, viz. :— [Se dun] = 0, . eae @.-. and comparing then (2) and (3), Os ov OM, deo Ai Sea ea pel shy been ea aun (7) | which could be regarded as physically obvious, as was done in the preliminary treatment (previous paper). so that We shall abbreviate (1) by writing m for = C for de,, 01 and M,G' for the “ general” terms; so that, pr = Dino ($u—Rlog il 0 MCs ; +3n.($.—Rlog +6 It will be convenient also to write Go’ for =o (M,G'), U1 the “general” terms in OWE. similarly G,’ for those in On 0 Ons Now, in the first place, we have to show that it is permis- sible to assume that (8) has already been so arranged that ° G', Gs’, Go’, etc., all vanish in the limit when © becomes , ete. Molecular Thermodynamics. 235 very small. This is essential to the rigour of the demon- stration that the Raoult-van’t Hoff laws are still the limiting laws of dilute solution when the solvent is complex. Now G’ is, at constant temperature aud pressure, a func- tion of ¢ 3, Cog) «.«+; a8 well as of ¢y, Co,..-.. Owing to the chemical equilibrium controlling cv, coz, ...., however, these quantities have, in the pure solvent, values depending only on temperature and pressure, and the departures from these limiting values, caused by the presence of solutes, will clearly decrease with the concentrations of those solutes. Thus as C diminishes, the ranges of variation to be considered of the variables co1, co, .... are progressively limited, the same being obvious in the case of ¢, cg, ..... But clearly any finite, continuous, differentiable function of several variables must behave as a linear function if the range of variation considered of every variable is sufficiently limited *. Thus Lt Geilo le . . E : . ° (3) C>0 0. Lt (MyG’) = Snoyloy + Vrrsls, 2 : : - (10) C>0 where Io), ...... Rin tie , depend only on temperature and pressure, being, in fact, limiting values of Go’, ....:. : 5 eee , respectively. But clearly (9, ...... a ape te , can be transferred to, and included in the “ linear ” terms,— in go, ..-.-. » di, re- spectively—whereupon the residual “ general” terms wiil . satisfy the requirement that G’, G,’, (Go,’, ete., should all vanish as C becomes very small. We shall assume that in (8) this adjustment has already been carried out. Returning now to (7) and comparing with (8), we see that Oy te fo eS ee eae, C>0 OM C>0 ul a = 4 | Seon os _ R log m cot) | = ou (say), ° (11) o>0 ; since, in the pure solvent, cq, ....-. , and ™ assume values dependent only on temperature and pressure. * Merely the obvious property of tangency in n-dimensions. The theorem quoted by Planck in treating simple solvents [‘Thermo- dynamics’ (Trans. Oge), 1917, p. 225] appears to be the particular case of this, when the ranges of variation of the variables are all located (as here in the case of ¢1, ¢,,...-) close to zere. 236 Mr. Bernard Cavanagh on Then we have oy = out | aon 3 a ov) eh ta ae but again remembering : 5 OY den =0, . oa 01 we get oy = =ou+ ( Xyi a(2*), ean (1.24) Sa. 0 and so from (8) : M ee : or =ou+ \e(— Rd log rae a\ + ie vei jel 0 % C=0 NC= =¢uth ac (1+ mO+ {; ScudGan. =~ (16) C=0 “C=0 Now (6) with (8) gives or) =o mC, / 17 (se Mo [ Sled P ae pe 1+ l+ mC 2 ue ( And if we write (16) in the feaedias : oe =o + R (- dlog (1 + mC) is Gu, 9 (18) C=0 it is at once clear that the “ general” terms thus adopted (and therefore, of course, the “ linear” terms similarly) are connected by the Gibbs fundamental relation [see note at end of this paper], for (MyG') being a function of 1q...... eee , homogeneous and of the first degree, Sy eG ed Ge ON WO: Mp2; €Go;’ + an, dG. =0, 1. é. M,dGy: +n, AGE =O" hoe ty Ce which means that [MoGu + 2nsGs'] or (say) M,G is, as a * The Gibbs relation might indeed have been used to obtain (16) direct from (17), dy, appearing as the integration constant. The above treat- ment [(12) to (16)] appeared, ihonree to be more interesting, and to introduce yy in a more natural and illuminating manner, in its relation to the original “ molecular ” expression for w. Molecular Thermodynamics. 237 function of Mg and nyng...... homogeneous and of the first degree. ‘And the same will hold for sihe ‘* linear ”’ terms, (if M,G) be accepted as the new “‘ general terms”), a cari of 0 I obvious importance since the “ linear” terms must alone remain when the solution is “ perfect.” ov retains the Ns simple form (17), which may now be written fo ae Mes oY = ($.—Rlog =a) RGA, Hs (20) since _/O(M,G)\ 3G . G=(Ss. Mo OG 2d) G being a function of cc, ...... only (besides T and p). Also, of course, s o(M, ') — J= SS GS a ° . e . hy G,=G aan (22) and MG = IM Ge srs ath et on (23) We observe, however, that this convenient arrangement involves theoretically a certain definite (though probably always very small) change in the division into “ linear” and “ general ” terms, since MG. = SarpiGor 2 rs Ge! : G! == ey, Sal Gee + Ye.G ==G a5 ( > Gio 2 en eel Bal Wee Wet ge (24) a c=0 Now it is readily shown that 0G’ 0G’ 0G’ ; G,=—— +m | G’—Se =~ + De, |; 25) ae oe gy Ge : Oe |e (25) and since, mop, being unity, Lmp dey, is always zero, we have ! G'—G= {> Oia By de oy ir 2G) O¢o1 C=0 To say that any such modification of the division into “ linear ” and “‘ general” terms is due to, and represents, a departure on the part of the solvent molecules from ‘“ perfect” behaviour would be to make a qualitative statement of no 238 | Mr. Bernard Cavanagh on practical significance. (24) is a quantitative statement, and the form to which it is reduced in (26) has practical meaning and value, as will be shown by means of a simple illustration in a short appendix to this section. From (26) it is clear that, owing to the relative smallness of the changes in ¢q, ¢p, ae , produced by the presence of solutes, the difference between the “ general” terms in the original “ molecular ” expression for wand those in the “ experimental ” expression we have obtained will generally be very smali. But it may have to be taken into account in making use of dynamical theory (at high concentrations). Of course, until and except when the “ general” terms can be given more definite form, we cannot say anything about the way in which they will depend cn the constitution of the solvent and its variation. For the present we have to suppose that Co1Co9 ..---- are eliminated in terms of T, p, ¢, ¢2, , from G, which takes some form GSR or 77 (cits <5. ) oa Sea Theoretically, and in the general case, the application of dynamical theory will precede this elimination of ¢9,¢9...... } will deal in fact with the original general terms M,)G’, so that, in greater or less degree, knowledge of the. constitution of the solvent and its variation will be necessary before such theory can join issue with practice, but in some cases, as the appendix will illustrate, this may not be necessary, even though the constitution of the solvent does affect the “ general” terms. It should be noticed that the value and convenience of our “experimental” expression for yis by no means entirely dependent on a deficiency of knowledge of the constitution of the solvent, though the latter makes it practically indispensable. In the “linear ” terms the effect of the constitution of the solvent is concentrated in the quantity m, the mean molecular weight of the solvent. In the pure solvent this will have a limiting value i, dependent only on temperature and pressure, and we can therefore regard the quantity (—R log mj) as included in the quantity $s, when our “‘ experimental” expression for w will finally take the form r= Mo | but R fie dlog (1. +m) | “o=0 c Bes (OER ete $3n[ e—B4 loges—log (7 +70 |] + Mot me (23) Molecular Thermodynamics. 239 when m is constant (at constant temperature and pressure) and therefore equal to mo we get R _— = My | éxc+ — log (J +m,C) | Mlo 4 +3ndd g.—R4 tog e,—iog (1+ 70) \] HONOR 0, (20) which is equation (52) of the first of these papers, from which the “second approximation ”’ equations were obtained. It is, of course, not possible to say how, in the most general case, m will depend on the concentrations of the various solutes, but an interesting case, of probably very wide application, may be treated and will at the same time serve as an illustration. This is the case where m can be written, with sufficient approximation, as a series of ascending integral powers of C, the total solute concentration. This can be shown to be the case, for instance, when the various solvent-molecular species behave as perfect solutes (in the true sense,—not in the sense of the Raoult-van’t Hoff laws). Some simple cases have been investigated, but the detail need not be given here. It will suffice to say that in the simplest case, for example where the solvent consists of two molecular species, mj and (2 mo), the one the doublet of the other, we find that ™ can be expressed as ifi= Tol 1 + (migC) + nCOC)?], where the values of @ and 7 depend, of course, on the proportions in which the two species are present in the pure solvent, but 7m any case cannot exceed % and 3/5 respectively (these maxima not being simultaneous). We may carry this expansion of m, which is formally convenient for our purpose, to one further term of which only the order of magnitude will matter, — =1+4 ATC) + n(imgC)? + EMC)... (30) 0 mm the last term being, as we shall find, altogether negligible if £ is no greater than about 7}. Approximating on the assumption that @ and » are of the | 240 Mr. Bernard Cavanagh on order of magnitude ;1,, we obtain : log (= +m) =aC0—JaC?+ 4,0", BI) = d log (1+ mC) =C(L—4a,C + 3ag0?—}a30%), (32) a0 where a, =m)(1—8), dg=mM,"(1—2(0—n) J, ° 5 ° (33) dz= my 1—3(0—+ &)] it being clear that, as stated above, & can be neglected altogether if no greater than about 74. We get then the equations | St Fart ROLL — 3a + Jaya] + Cry | ENS: Su (34) Aide = ¢,—R|[ log ¢, —aC + fay 7 —3a3C? | +Gs | \ and thence the successive stages :— | ov =$,, + RO[1—}ito(1—6) 0 -+.37%7C"] + Gy | TL. ge | (35) | ov =; — R[log ¢e;—77,(1—8)C + 4m,?C?] + G, | and | : SM =o ROLL 30") + Gye | Dae 2) es i =6;—R[loge,—mC]+G, | \ Sy ! and \ aa =$y+RC+ Gy | lay ; | ° e ° . . ° e (37) an, Tia log és +G, } ‘J. and II. being the first and second approximations obtained by the preliminary treatment in the previous paper. Taking } per cent. as the “ probable experimental error, Molecular Thermodynamues. 241 we get roughly the following upper limits (of total concen- tration C) for the applicability of the four successive stages of approximation : Approximation I.......... = | 55 el et 5 a 2M (38) een ee ers | A SO van ee 8 M. J —that is, considering aqueous solution, and assuming my to be about 40. The limits would be considerably different, par- ticularly in the case of III. and IV., if mp were given a very different value, as can readily be seen. The practical criterion of a perfect solute in a complex solvent is now that its behaviour should be expressed by that one of the above succesive approximations appropriate to the total-solute concentration of the solution, with the “* general” terms omitted. If the assumption of perfect behaviour in the case of a particular solute be made, an experimental determination of the quantities mp, 9, 7, ete., can be made and concordance in several such determinations made upon different solutes would tend to justify the assumption that perfect behaviour persisted up to the concentration at which concordance was found. According to the concentration reached (with concordance) some of the quantities my, 0, 7, etc. would then be known with some approximation (closest in 7m, next in 0, and so on). Then, on the assumption of “ perfect” behaviour on the part of the solvent-molecular species—that is, a sufficient approximation thereto,—these quantities would. suffice to discover something about the constitution of the solvent. Thus two solvent-molecular species would be completely determined by a knowledge of mp alone (that is, the propor- tions of the two kinds present would be determined),—and in this case the question of “ perfect”? behaviour of the two species would not enter. m, and @ wonld suffice to deter- mine completely three species, or would provide a check if only two species were present, and so on. The problem, however, as a practical problem is compli- cated by considerations which are the subject of a succeeding paper, viz. the question of solvation and partial solvation of solutes. Finally, it is proposed to consider one point with regard Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. R 242 Mr. Bernard Cavanagh on to the expansions of U and V, the total energy and volume of the solution. (Fuller consideration is postponed, as this paper is already rather long.) In “ perfect”’ solution we have U = Too, + Drsus 0101 | t (39) v= =N1Vo1 oh LNsVs and a) —— 791901 + 359s . . Q d . (40) where, according to the usage of the previous paper, Q is (U+pV), qs 18 (us + pus), etc., and we can write this Q = Mod co1901 + 3,5. . ¢ : A (41) >o1go1, however, depends (through ¢ 9, ¢.; ....) upon the concentrations of solutes present, but has, in the pure solvent, a limiting value gu depending only on T and p, and 201901 = au + 2Go1 | dey. eee 2 C=0 = gu + SqoiAcy. « 5... Seen And so Q == M,gu+ Snsde+ My dq Aco. <+ jae (44) Dilute now such a solution “infinitely” by adding a large mass M,’ of pure solvent at the same T and p, for which QQ’ aa M'gm. The united heat will be (Q ar Q’) aa (My 7F Mo gm 2 DNss a Mo>go1Acor: But since the solution is now “infinitely ” dilute, we shall have on bringing it to the original temperature and pressure Q’= (Mo+ My Jam + 2resg: = (Q + Q) — Mo Xo Aco. In other words, My3¢qo,:Aco, was the heat developed (evolved) on diluting the Xn, molecules of solute. That is, there is a heat of dilution of ih G2daAcn eh aiten item Ih Bak became (45) per gram-molecule of solute, in “perfect”? solution in a ““complex”’ solvent. The explanation of this apparent a ee gee Se Se awa! se ths Didi = Molecular Thermodynamics. 943 anomaly is that in a “complex” solvent the process of dilution is not quite so simple as in a “simple” solvent, being accompanied by a change in the constitution of the solvent—a reversion, in fact, to the constitution of the pure solvent. It is, of course, plain that MyXqo:Acoi, or ZquAnor, is the (“ chemical a} heat evolution accompanying this reversion (at constant temperature and pressure). With (44) analogously we have U = Moum 4 Srstte+ Mode, - . - (46) V = Moom + 2nsvs + ModvpAcn, - - - (47) and it may easily be verified from (L1) that Ur = m2 ae - (48) wuw= ph oom id Op. c APPENDIX TO SEcTion III. A simple example, which is essentially merely illustrative, but may possibly be something more, will serve to make clear the practical significance of equation (26) concerning the slight modification of the ‘‘ general ”’ terms. In section V. of the first of these papers was obtained for the general term in the case of a dilute solution of a binary strong electrolyte M,G'= RM,¢/e32, ... . ... (49) where g’ depended, in some way, upon a certain “ effective ” dielectric constant (D), which, at sufficient dilution, would be that of the pure solvent. Let us suppose that D depends on ¢, in such a way that ¢’ is a linear function of ¢, we — Ol -raeis)) a & . s (50) and also, in the first instance, that this effect of ce, on D is entirely independent of its effect on the constitution of the solvent, that is, that the slight change in the latter produced by the ‘addition of the electr olyte would alone [if produced, for instance, by some different solute, ¢,, the concentration R 2 244 Mr. Bernard Cavanagh on of the electrolyte being made quite small] have but negli- ible effect on D, then G/ is independent of the constitution of the solvent (practically), 2. e. (x8 aeu=0; . ae C=0 so that, eG one; -C + acs) } G, = @/ = 2Re'e7(l-t eee) es Gue= Bco1 Goi’ = —ERG,'c,°7(1 + 3ae,) } . If, on the other hand, we suppose that the effect of ¢, on D is entirely dependent on the change produced in the consti- tution of the solvent, and would be fully obtained if the latter could be brought about in some other way, while c, was made very small, then we should have > OG! Oen _ Res? oP | Oc Oa 06s (53) = Rostge | and ze l f {Se dey = Roe | cc toys eB) C=0 c,=90 so that Gia G 22d, oere, see (95) 2. é. in this second case, 2 I cone -| soe dey = Rdo’c2?(1+ sac.) C=0 G 2 = 8Rdy'c*(1+0e) |. (56) Gu = oi Gor’ =| S Den dey, = — 5 Rd 6°71 + 3aCs) C=0 In both cases, of course, Gs isl G,’, but they are not the same in the two cases because in the second case is a constant when 19, n2.... are held con- stant, but not in the first case. (for the same reaso >co1Goy’ is also different in the two cases.) In the second case we note that although the constitution Molecular Thermodynamics. 945 of the solvent enters into the ‘‘ general’? terms it does so only through the quantity D, and if this is the dielectric’ constant of the solution in bulk it can be measured and so determined as a function of c, without considering the corre- sponding variation in the constitution of the solvent, or the way in which the latter exerts its effect on D. If, as is probable at the less extreme dilutions, D is not the experimental or “‘ bulk” dielectric constant, but a certain statistical-average quantity of a peculiar kind, then its varia- tion is, at least partially, not due to a variation in the solvent, but directly to variation of c, as in the first case above. Clearly from the preceding it would theoretically be pos- sible in such a case to determine from comparison of theory with experiment whether the effect of c;on D was direct or indirect or in what proportion both, but it might not be practicable owing to the smallness of the effects to be measured, Notes ON THE “ Gipss FUNDAMENTAL RELATION.” Consider any property 7 of a homogeneous substance or phase, which is determined in magnitude by the composition of the phase and the quantity of substance considered. In view of the homogeneity it must then be proportional to the quantity of substance when the composition is fixed. Such properties are (at constant temperature and pressure) U and V, the total-energy and volume, Q or (U+ pV), which might be called the “‘reversible heat content,” and any thermodynamic potential such as entropy, free energy, Gibbs’ ‘‘ chemical potential,” or Planck’s yy, which may all be expressed (at constant temperature and pressure) as func- tions of the quantities M, M,.... of the constituents which suffice, under the conditions considered, to produce the phase. We can show, as Planck does in the case of y, that for any such property 7, ) T= sm, 07 ° . ° . ° (on) for if e be some infinitesimal fraction and we remove (ej) of the first constituent, clearly 7 is diminished by eM ar 1 Removing simultaneously the same fraction of the total quantity of each constituent we diminish 7z, in all, by Sei, SF, or e>M kom But, in so doing, we have 246 On Molecular Thermodynamics. simply removed the fraction e of the whole phase, without altering its composition, so that 7 must have diminished by err, that is whence (57) follows. When 7 is either Wy, or Gibbs’ “chemical potential,” whole system of ee in equilibrium can be congienee together, since then Sai , ete., are the same in every phase. 1 Without actually quoting Huler’s theorem, Planck remarks, in regard to ae that this relation means that vr, as a function of M, M,.. , ls homogeneous and of the first degree, though not, of course, in general linear, and the same remark applies to our typical property 7. From (57) we can at once get a more practically useful relation by differentiating both sides fully :— OT OT SME dM, = 257 aM, + 3M, a( or ‘ that is, OT =ud (37) = 0...) er Now equation (97) of Gibbs’ classical paper reduces at constant temperature and pressure to >m,du, = 0 and is then simply (58) applied to Gibbs’ ‘“ chemical potential.” (58) is therefore referred to in these papers as the “Gibbs fundamental relation,” but its general applicability to any property of the type of a (for a single homogeneous phase) is to be borne in mind. It is to be observed that while the constituents whose masses are M,M,.... must be sufficient to produce the phase under the conditions considered, they need not he all nepeesary——1hiey need not be the “ general-thermodynamic ” components. And since also (obviously from the form of (58)) M,M,.... need not be expressed in the same units we see that equally valid is the form End ($7) = 0, ee the “ molecular-thermodynamic” form, in which Ny Np « are the numbers present of the various molecular species. The Caleulation of Centrords. 247 It is an important point in the treatment of the two problems, “complex solvents” and “ solvation,” presented in this and a succeeding paper, that in the “ practical” or “experimental” expression for w finally obtained the “linear terms” by themselves satisfy the Gibbs fundamental relation, for in perfect solution these terms alone remain. And this is preserved in the successive approximations. The relation also serves as a useful check upon the correct- ness of the detail. Balliol College, March 1922. XVII. The Calculation of Centroids. By J. G. Gray, D.Sc., Cargill Professor of Applied Physics in the University of Glasgow *. | Bae position of the centroid of a plane are or area is usually determined by the application of one or other of the two theorems of Pappus. The methods described and illustrated below seem, however, to be novel ; they are useful in a great number of cases, including many to which the theorems of Pappus do not apply. Fig. 1. | Fig. 2. 0 A BP Consider a system made up of two masses M and m (fig. 1). Let the centroids of m and of the system M and m be at a and G respectively. Now suppose the mass m moved so that its centroid is brought to a’. G moves to G’, where GG! is parallel to aa’; and we have (M+m) GG’= maa’. As a first example, consider the case of a circular are AB mass per unit of length m (say). Let O be.the centre of the circle of which the are forms part. Now suppose the are * Communicated by the Author. 248 Prof. J. G. Gray on the rotated about O in its own plane through a small angle @, so that A is brought to A’ and B to B’. In effect the small portion AA! of the arc is transferred from one end of the are to the other. The mass of this element is mr, and it has been translated (virtually) through the distance 27 sin a/2, where « is the angle AOB. If G is the centroid of the arc, we have obviously mré 2r sin «/2 = mra OG 8, or : sake oe = 2r = [2 As a second example we take the case of a sector of a circle OADC (fig. 3). Let the sector be turned in its own plane through a small angle @ about an axis through O, so that A comes to A’, B to B’. The effect of the rotation has been (virtually) to transfer the triangle OAA' to OOC’. Fig. 3. Fig. 4. / The centroid G of the sector has moved parailel to gg through a distance OG 0. The mass of the sector is 47x0, and that of the triangle OAA' is 47°6c, where a is the mass of the sector per unit area. Since gq’= rr sing, we have 1 eee eae eee gP Ooarsing = 5 rac O0G8, ae 4 sin a/2 OG= 5 5 Again, let it be required to find the position of the centroid of a segment of a circle ABC (fig. 4). The segment is turned in its own plane, about O, through a small angle 0. A is thus brought to A’ and CV to ©’. If the mass per unit of area of the segment is o, the mass of the triangle DAA’ Calculation of Centroids. 249 is cOtrsindarsinda, or 4c0r*sin? ta. The area of the segment is $7’«—7? sin «cos 4a, and its mass is or*(4a—sin $e cos $e). Si rae eee ae ] ince gg’ is #7 sin $a, we have or (3a—sin $a cos $a)v0 = 2.067% sin’ 4a, or 2 vT=s 3 3%4—sin $a cos ta’ gin? L rsin® 5a where z is the distance of the centroid of the segment from O. Consider next the solids obtained by dividing a right circular cylinder into two parts by means of the plane abcd (fig. 5). Let it be required to find the position of the centroid of the lower solid. We suppose the solid rotated through a small angle @ about the axis OO! (the axis of | figure of the complete cylinder); ais thus brought to a’, bto b',ctoc’,andd tod’. In effect the wedge ebb'e'cc' has been removed from the solid and replaced in the position eaa'edd’. If AA denotes an element of area in abcd ata distance z from ee’, the volume swept out by this element due to the turning of the solid is AAaw@. The mass of this element of volume is pAA2@, and since the element is moved (virtually) through a distance 2, we have, if V is the volume of the solid, Le 7 VpO0GO = 298 > AAz’, where the summation is made over the complete area abed. Hence Vx OG = AK?, where A is the area abcd, and K is its radius of gyration 250 Prof. J. G. Gray on the about ee’. If « is the angle bOa, and / the length of the solid, we have for the sectional area 47°(27 —a) +7? sin $4 cos 4a. And since A = 2/rsin 4a, and K? = 47’ sin? 4a, we have Ge 2r sin? 4a a—ta+sin $a cos $a" Similarly, if G' is the centroid of the upper solid, we have oc $r sin’ ta—sin } ta cos 4 da! The positions of the centroids of the solids obtained by dividing a right circular cylinder into two portions by means of planes Oa, Ob (fig. 6) are easily determined. If G is the centroid of the larger solid, we obtain at once, by supposing Fig. 6. Fig. 7. -— -— ~ the solid turned through a small angle @ about the axis of figure of the complete “cylinder, so that a arrives at a’, and b at bY, Vx0OG = 2AK?sin de, where A is the area represented by Oa, and K is its radius of gyration about the axis of figure. We have, if / is the length of the cylinder, ‘_ 2lrdr’? sin $a his 47?(Qar —a)l 4 rsin ta 3, Oe. [f the cylinder is divided into two equal parts, we obtain Calculation of Centroids. 2901 the position of the centroid of each by putting «=7. Thus For the portion of a sphere shown by the firm lines of figure 7, we have, if OG is the distance of the centroid from the centre of the complete sphere, AK? 0G = = where A is the area abcd, and K is its radius of gyration about its diameter. Thus mr sin? a4? sin? da 4r°(1+ cos 4a) X 2rr 3 r Ty SE Gos ca OG = For the portion of a sphere enclosed by the firm lines of figure 8 we have 2 le gin L i Slt 2 Qa 3 wrgin da ~& Yor—a where wz is the distance of the centroid from OO’. For a hemisphere «=77, and 5 = 8 / ae Finally, for the portion of an anchor ring shown in figure 9 we have, if ris the radius of the ring and a the radius of the section, Gos: ma? (ta? +77) 2 sin ta 2a —o *° Ta xX 2arx 2 _ 2(7r?+ 4a’) sin $a r (27 — a) : which reduces to 2r/m when «= and a is very small in comparison with 7. If a body is floating partly immersed in a liquid, the 252 Messrs. Trivelli and Righter on Silberstein’ s distance of the centre of buoyancy B (the centroid of the displaced liquid) from the metacentre M is given by BM = AK’/V, where V is the volume of the displaced liquid and AK? is the moment of inertia of the water-line area about the intersection of the wedges of emersion and im- mersion. The equilibrium of the floating body is stable if Fig. 9. M is above G, the centroid of the floating body, that is if BM> BG; it is unstable if BM m, and satisfy the general recurrency formula Omi =Am—1,i-1 + Wri, iy which enables us to write down successively without trouble any number of them. Thus, up to m=10, we have the following table which the reader may continue to extend at his leisure. Columns correspond to con- stant m, and rows to constant 2. 1 1 E iE 1 1 i 1 it 1 0 1 3 ii 15 ol 63 127 259 511 0 1 65-525 90 301 9686 3025 9330 0 iL 10 65 350 1701 7770 34105 0 i 15 140 2401 138706 76300 0 i 21 266 3997 37688 0 1 28 462 7231 0 1 oo, ':):) FeO 0 1 45 0 ] We way mention in passing that any am; can be represented by + imi | o— (1) @—D™+ (5) G—2ym—......+ (3) J. But for any numerical applications the table will be found more convenient. * In a large number of trials of the same experiment. + Cf. e. g. Lehrbuch d. Kombinatorik by E. Netto, Leipsig, 1901, p. 170. 262 Dr. L. Silberstein on a Quantum By formula (5) we have for instance the probability of hitting but one grain 1 Aye Nm-1? which is an obvious result. The probability of hitting two grains will be pn 2) =e — a yee which is (approximately) 2”V times as largeas the preceding probability, and so on. If m be kept constant, pn»(i) will increase with growing 7 up to its largest value for 1=m, if m=, or fori=N, if m> NV. Butit would be futile to expect, on an average, that distribution to which corresponds the greatest probability *. For all other distributions have some generally non-negligible probabilities and these are by no means symmetrically spaced with respect to the largest one. The only reasonable way of determining the number k is to define it as the average of 7 taken over a large number M of trials. Out of these M trials a number M »p,,(2) f of trials will give each i grains affected, and the total number of grains affected in all M trials will be 2Mzpn(2), to be summed fromz=1 toz=m, if m= Nand tor=Nif m> NV; but since a,,=0 fori >m, we can as well extend the sum in each case from 1=1 toi= J. | Dividing this sum by M we shall have the average number of grains hit in one trial, 7. e., by (5) INA N ean = nae ao oe (6) This, with m=Nan, is the required rigorous formula for the number of grains affected, 7. e., hit once or more. In order to see how this complicated formula degenerates into (4), which, of course, will be our working formula, develop the sum in (6). Collecting the terms in V4, WV? etc., and taking account of the values of ami, it will be found that ee a 1 es (- ls jie ees N 2 iP WN? 3 —.eet N y) * Tf, say, m=, the most probable distribution is the equipartition as m) ! would give as the number of grains affected k=m=WNna, or just the first term in the series development of (4), which would be hopelessly wrong unless m WN were very small. + With a deviation ruled by Bernoulli’s law. ke (a quantum per grain), corresponding to pm(m)= Re Theory of Photographic Exposure. 263 or, dividing by WN and subtracting from unity, k m 1 m 1\n 1~ wait (T)(—H) +--+ CR) (— 3)” 2. €. ultimately hk £4 1 \m F fu = (1%) hy ee ama This is rigorously equivalent to or identical with (6) for any mand N. Now, for large N, and any m, equation (6’) in which (1—1/N)*=1/e, asymptotically, gives at once k _m ee —l—e W=1—e7™, This is the connexion between the rigorous arithmetical formula and the exponential one. It is needless to insist that under the conditions prevailing in all practicable experimental cases there is more than sufi- cient mathematical accuracy in formula (4) h=N(1—e-"), which, apart from minor modifications, will henceforth be used in what follows. | This formula is of the familiar type proposed (1893) by Elder, with the notable difference, however, that while his exponent contained a free “‘ parameter ” or coefficient to be evaluated empirically and principally depending upon “ grain- sensitivity ’’ and wave-length, both coefficients in (4) are completely determined, and the exponent moreover shows an explicit and most essential dependence upon the size (a) of the grain, and in the right sense too, 7.¢., giving an increase of the “speed” with grain size. The comparison with experimental facts of Elder’s and of a number of other formulz, constructed empirically, is too well-known to be discussed here *. Suffice it to say that, although it repre- sents to a certain extent the phetographic behaviour (the “characteristic ” curve) of some emulsions, and particularly those with what is termed an “extended toe,” it cer- tainly shows considerable deviations from the observed characteristic curves. Yet it will not be forgotten that all these comparisons bore upon the resultant total photographic densities, containing or integrating the effects of grains belonging to a broad range of sizes (a), instead of equal * Cf. for instance, a paper by Dr. F. E. Ross, Journ. Opt. Soc. Amer. vol. iv. p. 255 (1920). 264 Dr. L. Silberstein on a Quantum grains, so that no better agreement could be expected. The refined experimental tests which are now in progress in this laboratory, and by means of which it is hoped to corroborate the proposed theory, deal, as they should, with separate size-classes of grains. 7 But questions concerning the comparison of the theory with experiment will be treated in a later part of the present and in subsequent papers. 4. Passing next te the case of an emulsion of any type f(a), it can be easily proved that the approximate formula (4) will hold for each class of grains separately. In order to see this it is enough to consider the case of two distinct classes of grains. Thus, let there be NN, grains of size ay and WV, grains of size a, spread over the (unit) area S of the plate, and let n light-quanta be thrown upon S. Of these a number m=nNja, will fall on the a,-grains and a number mz=nNoa, upon the dg-grains. It remains only to be found how many a,-grains will be hit by the m, quanta, and how many of the a,-grains will be hit by the m, quanta. Now each of these is a problem of the kind we have already treated. The number /, of a,-grains hit will be given by 1 eee Dc ee ea eee and similarly for the a -grains, so that k, and k, will each be determined by the previous formula for & with JV, a replaced by Yj, a1, and Ng, ag respectively. Similarly for an emulsion consisting of three or more classes of grains. Thus, also, for an emulsion of any type f(a), the number of grains ranging from a to a+da hit and (if p=1) affected by n impinging light-quanta will be 4, da, where, apart from the correction € kg=((a)[t—e-"). 6 er The total area * of silver halide affected or made developable will be found by extending the integral K-= akede gE faa: 30 as over the whole range of sizes, say from a, to dy. If, for instance, f(a)=Ce-**, say from a=ay, to a2, where C, » are constants, as in the case of some films and plates = NA, * It will be kept in mind that a stands for the “ efficient ” area of a grain (plate), 7.¢. for the orthogonal projection of the grain upon the film base. Theory of Photographie Exposure. 265 investigated in this laboratory for their frequency curves, then, with A written for the total area of silver halide, K=A— eos { fa + (n+ p)a,le7# tm (n+)? —[1+(n+p)a,je"H+}, But this only by way of illustration. The fundamentally important thing will be formula (7), applicable to each size-class of grains separately. In fact the experimental verification of the theory now in progress in this laboratory deals, not somuch with X but, as it should, microscopically, with k=k, for each class of grains separately, including clumps of grains. Before passing to a further discussion and development of the elementary formula (7), but one more remark concerning the presence of more than one layer of grains. The case of two or more layers will at once be seen to be reducible to that of a single layer. In fact, either a grain of, say, the second layer and of size a is not shielded by any of the first layer or else it is thus screened off and only a part 6 of it remains uncovered. In the former case the grain in question will simply be classified among those of size a of. the first layer, and in the latter case among those of size b. This will hold with respect to the exposure to the impinging light- quanta, and 6 will also be the contribution of the grain in question to the photographic density; for its covered part will remain inoperative. Similarly for three or more layers. In fine, the presence of a plurality of layers of grains will modify only the frequency curve N,=/(a) which would otherwise belong to a single layer. We shall henceforth assume that this factor has already been taken into account in constructing the function f(a) or in microscopic counts of the grains within every particular size-class. We disregard bere, of course, such factors as a possible absorption of light in additional strata of gelatine. | 5. Dependence upon wave-length—Once more return to the elementary formula (7) or (4). Denote by s the ex- ponent so that == |] —eg- 5, N Under the more or less implicit assumption that the trans- versal dimensions of a light-quantum are negligible in comparison to those of a grain, we had s=na. But it will 266 Dr. L. Silberstein on a Quantum be seen presently that such an assumption is too narrow and unnecessarily so. In fact, substituting the number n of light-auauie from equation (1), the exponent s* will become s=7 ED, Or, it we put for brevity s= a/he, oi eer Ng Sacra ea (9) which may be considered as a constant, S= BE. ON.) i.e eae Here F is the incident light energy (exposure) and A the wave-length of the light assumed to be monochromatic. | Thus the sensitivity exponent would be directly proportional to the wave-length, and the number & would, for constant Fi and a, increase steadily with the wave-length of the incident light up to the photoelectric critical value A, and then drop suddenty to zero, i=l cae QS xX < oe k ==(() N> Ne Now, such a sensitivity curve does not seem to resemble the familiar experimental sensitivity curves which show a more or less gentle maximum followed by a gradual decrease down to zero. It is true that such experimental curves Tf represent the resultant effect due to grains of a whole range of sizes, so that the k—) curve belonging to a single class of (equal) grains may well be of the said abrupt type,—a question to be decided only by micro-spectrographic experi- ments and counts now in progress. Yet it seems advisable even at this stage to provide for the possibility of smooth maxima preceding the critical wave-length 2.. This might be obtained by attributing to grains of different sizes different values of X. For then, although the curve of each grain class would end abruptly, the superposition of such curves ending over a range of different abscissee, might properly displace and smooth out the resultant maximum. The correctness of such an assumption (A, a function of a) can at any rate be tested by direct experiments f. * s/H can be referred to as the “ sensitivity exponent.” ' + Apart from the fact that they are not taken for H=const. { Preparations for such experiments are now being made 1 in this laboratory. Theory of Photographic Exposure. 267 Another way is to take account of the possibly finite trans- versal dimensions of the light parcels, which may perhaps be comparable with those of the lesser silver halide grains. Let us assume, therefore, that a light-quantum, of suffi- ciently high frequency, becomes efficient in affecting a grain only if it strikes it fully, or almost so. To fix the ideas, let r be the equivalent radius of a grain, 7. e., such that =r", and similarly let p be the average radius of the transversal section of a light parcel (so to call the space occupied by a quantum of energy). Then the efficient area of a grain to be substituted instead of a, will be 2 a’ =1(r—p,/?=a E e A es and we shall have for the exponent s, instead of (10), s=na =6 La | 4 x. Longo 7, See (10 a) amd s—() for p> 7 or A> Xe. It remains to assume in a general way that p, which may be the average of section-radii different even for parcels of the same wave-length, is itself a function of the wave-length increasing with A, without prejudicing, however, the parti- cular form of this function. Certain easily ascertainable broad features of such a function and thence also of the resulting factor in s, poryarn[1—P]",. Portia BP Ghai) will suffice to ensure a maximum of the sensitivity exponent between X=0 and A=A.. The value of A, itself may still turn out to depend on the size a of the grain and on its physical conditions as well. Every process which will make the liberation of a photoelectron trom the grain (crystal) easier will lengthen A,. Part of the effect of sensitizing may arise in this manner. But questions of this kind must necessarily be postponed until some further experimental data are gathered. Of such a kind is also the question whether p (which, for a given X, may also extend over a whole range of values) attains at all the semi-diameter 7 of even the smallest of the actual grains, and whether the corresponding wave-length , entailing the vanishing of s, exceeds or is smaller than 2, as derived from direct photo- electric experiments. In absence of all knowledge concerning 268 Dr, L. Silberstein on a Quantum the spatial properties of light-quanta it would be utterly unjustified either to deny or to assert that their lateral dimensions are at all comparable with those of a silver halide grain (of the order of one-tenth up to several microns) *. If, by way of illustration only, p is propor- tional to a power of A, say p=dA*, the only condition for the existence of a maximum of ¢(Q), and therefore of sensitivity, wil easily be found to be «>0. If this be satisfied, the maximum will occur at a wave-length 2,, given by (2e+1)brA, =" increasing with the diameter of the grain and bearing to Ao the fixed ratio Xm _ Ve x == Zea ue Asa matter of fact, the maximum sensitivity is known to shift (by two or three hundred A.U.) towards the red by making the grain coarser. But thus far too little is known of the quantitative aspect of such an effect to entitle one to con- sider the above equation as anything more than an illustrative example. The precise form of the function p=p({d) can only be derived from spectrographic experiments followed by microscopic grain counts, or if arrived at by a guess, has to be verified by them. Such experiments are now in preparation in this laboratory, and their results will be reported in due time. A shift of the maximum sensitivity towards the red or the infra-red ean, of course, be brought about by a function form more general than a mere positive power of the wave-length. 6. Generalities, and preliminary account of experimental — tests.—The chief and most immediate consequence of the proposed theory is the essential dependence of the propor- tionate number of grains affected, k/V, on the size a of the grain, viz., the rapid increase of the former and, therefore, of “the speed ” of an emulsion with the latter. Now, it has been known for a long time that (cvteris paribus) the speed increases notably with the size of the grain, and we shall see from the experiments to be described presently to how * According to E. Marx, Annalen der Physik, xli. pp. 161-190 (1918), the volume of a light-parcel, which according to him is only a ‘“‘ concen- tration place ” within a continuous distribution of energy, is proportional to 4 and amounts for D-light to almost 8.1077 cm.*, which even with a leneth of 10 cm. (200,000 D-waves) would still give a section area 8.107 3 cm.?, just of the order of about the average grain area. ‘here is, of course, nothing cogent about Marx’s estimate, yet the matter is not without interest. Theory of Photographic Hxposure. 269 large an extent this is actually the case. But perhaps the most tangible proof of the essential correctness of the assumption of spatially discrete as against continuous action”, seems to be the mere fact, disclosed by microscopic counts, that out of a number of apparently equal grains subjected to a sufficiently weak exposure one or two are affected while the others, nay their next neighbours, remain perfectly intact. It would be in vain to ascribe to these survivors a greater immunity or indifference to light. For it is enough to protract the exposure a little to make them succumb in their turn. Now such a behaviour is most typical of rain as contrasted with flood action, and the discrete light-quanta, hitting now this and now that grain, appear to be a most natural inference, while all attempts to bring into play the individual “sensitiveness” of the units seem to involve considerable difficulties. As to the dependence of the number of grains affected upon the wave-length, little more of interest in the present connexion is known than the qualitative fact of a shilt towards the red of the maximum sensitivity with increasing size of the grain. Moreover, the available curves repre- senting the sensitivity across the spectrum concern the emulsion as a whole and not the separate a-classes of grain with which we are primarily concerned. Spectrographic and spectrophotometric experiments of such a kind, to be aided by direct photo-electric measurements, are now in progress in this laboratory, and all discussions involving wave-length will best be postponed until the results of these experiments and of laborious microscopic counts are forthcoming. Before passing to the mentioned quantitative test of the dependence on size, but one more general remark. ‘The reader will have noticed the complete absence of the time- variable in all our formule, the exposure entering only through the total number n of light-quanta or through the energy / which, in obvious symbols, is (2 dt. The pro- posed theory, therefore, as thus far developed, does not take any account of the little infringements against the reciprocity law t, in short, of the so-called “failure of the reciprocity law.” Now, it is by no means my intention to deny the * A rain as against a flood, of light, that is. t+ This early law asserted the dependence of the photographie effect (density) upon J and ¢ only through the total incident energy or ex- posure { Lat. For constant intensity this is Zt, whence the name of the law, relating to intensity and exposure-time as factors of a constant product. 270 Dr. L. Silberstein on a Quantum reality of these infringements which have been extensively studied and condensed into empirical formule by Abney, Schwarzschild, Kron and others. But it has seemed inad- visable to encumber the very beginnings of the proposed theory by complicated details of such a kind*. The failure of the reciprocity law can more profitably be taken up later on, after the fundamentals of the theory have been somewhat solidified and extensively tested, and the prospects of mastering the “failure’’ theoretically are by no means averse, a very promising scheme seeming to lie in the possibility (suggested by Joly and taken up by H. 8S. Allen) of the liberated photo-electrons being regained by some of the grains which were deprived of them by previous impacts of light-quanta. In fine, the failure of the reciprocity law as well as the facts known under the head of “reversal” have at first to be neglected and considered as future problems for the light-quantum theory united with Joly’s photo-electric theory, problems to which these com- bined theories seem well equal. To pass to numerical facts, a short description will now be given of the results of certain experiments undertaken in this laboratory by A. P. H. Trivelli and Lester Righter f which seem to corroborate the proposed theory most emphatically. In order to have a much wider range of sizes a than is usually afforded by the single grains, Trivelli and Righter applied their counts and area measure- ments to clumps of from one to as many as 33 grains, basing themselves upon the well-supported assumption that if one of the component grains be affected, the whole clump is made developable. (This, at any rate, is the behaviour * R. E. Slade and G. I. Higson, “‘ Photochemical Investigations of the Photographic Plate,” Proc. Roy. Soc. xeviii. pp. 154-170 (1920), on the contrary, make the failure of the reciprocity law their point of departure. They mention at the very beginning (p. 156) the possibility of a light- quantum theory and write the Elder-type of formula in J, ¢, remarking even that its coefficient would have a different value for each size of erain, but being discouraged by or rather preoccupied with the failure of the reciprocity law do not enter into the details of the probability problem, which would have disclosed them the structure of that co- efficient, and without much ado dismiss the quantum theory as “impossible.” Independently of Slade and Higson the possibility of a discrete theory (radiation in “ filaments”) is mentioned by I. EK. Ross, Astrophys. Journ. vol. lii. p. 95 (1920). Dr. Ross, without being prejudiced against such a theory, notes even that it would lead ration- ally to a mass-law, but does not enter into the details of the probability problem and does not develop the theory. + For technical details of these laborious experiments, see Trivelli and Righter’s own note in this issue of the Phil. Mag. p. 252. Theory of Photographic Exposure. 271 of the larger, flat grains piled upon each other in part, although the smaller, spherical grains, in less intimate contact, may perhaps behave differently.) Such being the ease, their experimental results should be covered by our formula with a written for the area of the whole clump, no matter how large and how numerous its components. This has seemed a rather severe test but the more so tempting and instructive. Since all the classes of clumps were given, in each trial, a unique exposure (through a blue filter specified loc. cit.) and there was no question of varying 2, it will be most convenient to retain in the corresponding formula the original light- quantum number n as the parameter common io all clumps. Thus the formula to be tested becomes h 2 | eal =nal =na(1— p ; or somewhat more conveniently for computations, if c=’ be the (average) area of the transversal section of a light parcel, k a}? log (1-4) =-ra [14/2] . eis (12) In the following table the first column gives the number of grains in a clump, the second the average area a of a clump in square microns, and the third column the per- centage of clumps affected out of all (JV) clumps of each kind originally present, 7. e. 100 & Tee eRe 2 as deduced by Trivelli and Righter from their observations, Clumps of @ in p?, Cone. Oeics Ay. PRI GEMIND o.cscsec.-6 0-754 165 16°2 +03 foeraims, fhcti.... 1-925 44:9 48°4 —3'5 (2 ae eee 3°03 76:6 68:9 +8°3 Ca ee A'88 8771 87°3 —02 yl eee 6-18 96-0 93°3 27 By Bee Be ee Ne 742 98°2 96-4 1:8 OSS A is seae (8°6) 100 98:0 2°0 = alpeitiey RSA (9°) 100 99:6 0-4 Des eet grt’) 100 99°38 Or2 ‘pa ah Tet (12° ) 100 100 0:0 ete., ete, etc., ete. idem, idem. idem. A PTAA | cope oan >24 100 100 0:0 25 100 100 0:0 The most reliable a-values are those for the clumps of one and of two grains, being averages of the largest numbers 272 Dr. L. Silberstein on a Quantum of individual clumps ; the following a’s are vradually less reliable ; from the 7-grain clumps onward the areas (bracketed) are only extrapolated, but since here y has practically reached 100, no greater accuracy is needed. The fourth column contains the theoretical values of y following from formula (12) with the constants, determined from observations 1, 2, 4, n=0°5724 per a c= 0091 ae or p=0'176u. The fifth eolumn gives Ay= ose venee The agreement is certainly very pronounced, the differences being, perhaps with the exception of the third, well within the limits of experimental error chiefly in the a-estimates. The fitting could be made even closer by retouching the constants n, a, but this is scarcely worth while, the formula itself being of a statistical nature, and the agreement being good enough as it stands. The same is manifest also from the figure giving a graphical representation of the last columns of the table. (12 a) | 1001 E og 2 0 2282008 8-89 8-8 88-8 8 5-886 OO 6 i : YU 60) — CALCULATED | 4 e OBSERVED 49 3 26 8 5 10 15 20 25 a The reader might think that the finite section o, or whatever this parameter may stand for, has been forced upon the light-quantum and that the observations might perhaps be as well represented with e=0, and another value of n. But actually, just the contrary has been the case, inasmuch as the author first tried the simpler formula log (1—4/N )=—na, and then only found himself compelled to take in the correction factor as given in (12). In fact, Theory of Photographic Exposure. 273 dividing the observed values of log (L—4/NV) by the areas the reader will find that the quotient increases considerably and systematically, apart from a casual drop at the fourth clump, throughout the whole series of the clumps. Thus, the correction factor seems to come in quite spontaneously. On the other hand, there is nothing unlikely about the values of either of the constants (12a). Our units of area being here square microns, we should have 57 millions light- quanta per cm.” (about which judgment has to be suspended until absolute energy measurements are available), and as the cross section of the space occupied by each of them (on an average) a little less than one-tenth of a square micron or a diameter of about 0°35 micron. Since each is presumably of about the order of a million wave-length long, they are rather slender at that cross section, and, instead of light parcels, as they were called above, would perhaps deserve rather the name of light darts. In Hinstein’s own theory there is nothing on which to base an estimate of the volume occupied by a light-quantum, but on Marx’s less radical views this is about 8.10~-‘ em.® for D-light and proportional to A4, and therefore in our case (narrow blue spectrum region with maximum at A=0°470u) about 3.10-* em.’, which with the said cross section would mean a length of 3.10% or over six million wave-lengths. But this by the way only. The important thing is to see whether the above numerical value of the average cross section of blue light darts will continue to fulfil its function with regard to the remaining “‘steps’? (weaker and weaker exposures) of plates coated with the same emulsion, the above being the highest “step.” These have just been completed in this laboratory, ceteris paribus with the above one, and are now being subjected to counts and area measurements. ‘This material will also serve to test the constancy of k/N if, varying n and a’, their product is kept constant. An account of the results of these and of several other experiments now in progress will be given in future papers. I gladly take this opportunity to express my best thanks to Dr. C. i. K. Mees for having proposed to me the problem of “discriminating, if possible, between the consequences of a diserete and a continuous exposure theory,” and to my col- leagues Trivelli and Righter for furnishing the results of their experiments. Rochester, N.Y., January 19, 1922. Phil. Mag. 8. 6. Vol. 44. No. 259. July 1922. dh ya XX. An Analytical Discrimination of Elastic Stresses in an Isotropic Body. By Rh. F. Gwytuer, M.A.* Gy G. B, Ary obtained from mechanical considerations (British Association Reports, Cambridge, 1862) a solu- tion in Cartesian coordinates of the mechanical stress- equations, but he ignored all elastic requirements. In one sense this paper may be regarded as an extension of Airy’s scheme, though it has nothing in common with that scheme either in general plan or detail. The method is purely analytical, depending upon general solutions of the mechanical stress-equations and upon the development of a scheme for the selection of those stress- systems which satisfy the stress-strain relations, briefly called Hooke’s law, from the general mechanical stress- systems. It is shown that the elements of a mechanical stress depend upon an arbitrary primary stress-system, and, to form a connexion with the stress-strain relations, I introduce a subsidiary, but allied, stress-system which is such that the vector system naturally deduced from it possesses inherent qualities distinctive of the displacement conrespondie to an elastic stress-system. The main body of the paper consists in dhysteene the requirements necessary to ensure that the stress-system should be an elastic stress-system. The displacement, according to this method, becomes somewhat incidental, however necessary, and the elements of stress are given prominence. ‘There are no displacement equations. “In the first instance I deal with a body under tractions only, and extend the scope of the results later. I. INTRODUCTION. 1. By first treating certain ancillary matters as lemmas, the steps in the final stages can progress more steadily. The mechanical equations of stress in a body under * Communicated by the Author. Analytical Discrimination of Elastic Stresses. 275 tractions only are, in Cartesian coordinates, Obeseu - ho Oey eee Ot Saar Cee aU 3, 38 _ Sc a or oS. oR — SSE) ie nce ee and they are identically satisfied by the values be 0° 0°; | 9 OV a7 a2 Oy? **ayoe Q=- 078) 0°03 ae Cae iets te 037 Oxo2 0°O, 078; Obs + eee Oa +30 oa ow Oh ola" Oy 0¢ Ox" Ovoy Or02 pa 020, * Ob; 4 or; ai O23 0202 ez0z 8OY 0y02’ 070s Oy O's Ordy Oxdz OyYdZ O27" These contain six arbitrary an1 general functions, and form the general solution of (1). Lemma A. These six arbitrary functions have the same mode of resolution on transformation of axes as the elements of stress. For proof I use the method employed in the subject of differential-invariants. Imagine the axes of coordinates to be rotated about their own positions by the infinitesimal amounts @,, w,, w,, and consider the consequent changes in whatever quantities we may be considering—components of a vector, elements of a stress, etc. These changes will be linear functions of Wr, Wy, Wz. T 2 276 Mr. R. F. Gwyther on an Analytical Discrimination For example, in the case of the components of a vector u, v; w, the changes will be : aa —Oyw+ozv, in’ v, —O,U+o,W, In WwW, —@,0+@y,l. These changes will now be represented by partial dif- ferential-operators ©,, Q,, QO; arranged to produce the ~ coefficients of @,, @, @, in the quantities considered respectively. _ Thus, for vectors generally, aL fo) QO, ee a Oo os we, S NOs a ee Sie This covers direction-cosines (/, m,n), and may be made the basis of most operators. For example, we may deduce for stresses generally : 40) if r) Oe 28(s—-sn)+®-O SUS 47 9, oO R Vian eintig aig aes 2, = 21(S.—yp)+@-R) sy aU oe aie | OQ, oT UES Ls tbe! ce. On e QO, 20 ( a oO P) sagt Se The differential coefficients ce oO ce) resolve as com- O® OYy102 5) ponents of a vector, but, for simplicity, we must introduce a different notation. — al Write d; for 0/O2, d, for 0/dy, ds for 3/dz, and d,, for Ms / 9 ky o 11 TOT 07/02", di. for 07/Ox0y, doo for 07/dy?, and so on; then for first differential coefficients ao fe) O, = Ti ge Od,’ etc., of Elastic Stresses in an Isotropic Body. 277 as for vectors, and we deduce for second differential co- efficients PMO TMs NOU pig Deus e 3 D1 = dis dia sr, +2 (So 5g) 7 (dose) at ‘cede 3 ee eae. aga OO Sa, + 2d,3 Soe | (d33 dis a 0 fo fe 0 0 QO; 7 dos => Odis — Oa Odo a7 2d» (.5--9-}- (dy1— dep) Ody," It is required to establish that in (2) 6;, 0, @3, Wi, ro, Ws act as stresses. Actually I shall assume that the operators QO), Qe, 3; act upon them as upon the elements of stress, and first examine this hypothesis when the three operators separately act on the six equalities in (2). Selecting the first equality in (2) and applying the operator ©, on each of the two sides of the equality, then Oe = ®) 3 and 20s eo 0°63 tonal ROY ey etal ONC ees dal 1 32 O, PY ots “Oy Oz becomes Or 5 ON , 5 07(Os—F2) 9 0° (0,—63) 9 2 ate ay + 3: RE On. EiKg (3 -32) my which is null, and the hypothesis is not negated. In fact, the hypothesis is not negated in any case. The argument is then as follows :—If we had written 040, 028, ete. in the equations, and had proceeded to find these 18 quantities, we should have 18 linear equations from which to find them. The solution is therefore unique, and cannot differ from that employed by hypothesis and found to satisfy the 18 tests. We shall therefore regard 6, 02, @3, Wi, Wo, Wr3 as acting as elements of a stress, though they have not the proper dimensions. It is proposed to find the conditions which must exist between these primary arbitrary stresses in order that the elements P, Q, R, 8, T, U may be elements of an elastic stress. 278 Mr. R. F. Gwyther on an Analytical Discrimination Note.—If, in the ordinary notation for strains, we give a, b,c each one-half of the usual value given to it, strains would follow the same laws of composition and resolution as stresses, and would therefore have the same differential operators. In this paper, I shall use a, b, ¢ in this sense— that is, one-half of their usual.value. Lemma B. Hxcept when we have reasons for keeping the expressions quite general, it will suffice to limit the arbitrary stress- system to such stresses as have the co-ordinate axes as their ‘principal axes. If in equation (2) the elements of stress, P, Q, ete., are made zero, the set of equations will then be recognized as indicating that 0,=e, 06,.=f, O:=9, Wi=a, We=), W3=6, where e, f, g, etc. are elements of strain arising from some arbitrary displacement. Hence on the right-hand side of equations (2) we may always replace 0, by 0;—e, by 0.—f, 03 by 6;—9, Wi by Via, Wo by we—b, 3 by W3—¢. Consequently we may eliminate several sets of three functions, such as Wi, We. and 3, when some displacement is possible which makes, say, Wi=a, Wr2=, W3=e. Hence which is the form given by Airy, is a quite general form of solution, although for the purpose of this paper the full form given in (2) is requisite until we have decided upon some particular set of axes. 2. The choice of a vector to represent the displacement, and the descriptive criterion of elastic stress. The mechanical stress has been represented in terms of an arbitrary stress-system, and it is possible and desirable to represent the displacement in terms of a similar stress- system. For this purpose I form a subsidiary stress-system, of Elastic Stresses in an Isotropic Body. 279 indicated by 0, eS O,-+ Hinail Eee 0, 4- EAL —wW, oe ro, — 3. This subsidiary system may be described as comple- mentary to the primary stress-system, in the sense that the two together form a hydrostatic pressure whose intensity is one-half of the sum of the principal stresses or one-half of the First Invariant of the primary stress-system. I shall form the assumed components of the displacement from the elements of this subsidiary stress in the manner of forming a force-system from a stress-system. Thus, I shall write fe Fee: 2nu = © (02+8;— 6) 2 022 OY Oz’ ESpoes |, @ howe Sea 2nv =—2 Sener (0;+ 0; 0.) 2 55° ab 9g Ove toh. 0. 2nw= —?2 aye 2 Oy oe (0,+ 6,— 3), e 6 (3) Forming the values of 8, T, U from these on the elastic stress-strain hypothesis we have g — 01 Oh _ O's O_O" OY 02: .Otoy) Or 0e Oy (102 ete. On equating these to the values for the same elements given in (2), we find they require Vi = 9, Vr2=0, V3=0. . . (4) Since {0), 42, 03, Wi, Wa, Ws} act on transformation of coordinates as elements of stress, it follows that the system must consist of a hydrostatic pressure and a general stress-system, each of the elements of which is a Spherical Harmonic. That is, A=P+xX, B=h+X, O=h+Xs V?X1 = 0, V Xe — 0, VX == ae (5) This is the descriptive criterion of an elastic stress- system. where 280 Mr. R. F. Gwyther on an Analytical Discrimination 3. Completion of the discrimination. The metric criterion. The remaining requirements of the stress-strain relations may be written Ne OY Ov Ow Ow P+Q+R= (3m—n) ($ page Sy tae) «2k eae) which are to be completed from the values in (2) and (8). The first two only confirm the descriptive criteria of (4) and (5). The last leads to 2 2 —V?(91+02+ 03) + {<% eres +S Oy O's ohne ee i O21 10105 One, —" 1 V(8: +0+85)— ae ae ot S38 Ov , 9 O's | 9 Obs Poo one 2seanh ==" —n and therefore to (3m+n)V7(O; + 0, + 85) 0°6; a O70; 5 OW Oe | ON = 6m mS Se ee a oe or (m+n) V : ae 0x1 , O'X2_, O'X3 Oy oO dome 9 OVS = 2m Da? 4 5 By? + 523 erry ae i, (7) Owde "Ow oy which is the metric criterion and completes the dis- crimination sought for. This can be integrated, and gives @ in terms of the y’s and w’s, of which we can always take the w’s to be null, when desirable. This completes the investigation for Cartesian coordinates under normal tractions only and with no inertia terms. With the conditions in (4), (5), and (6), the equations (2) give the general elastic stress-system, and these con- ditions discriminate an elastic stress-system from any other mechanical stress-system. of Elastic Stresses in an Isotropic Body. 281 4, The inclusion of inertia terms. We must now modify equations (1) by writing pii, pv, pw on the right-hand side, where wu, v, w are to have the values given in (3). We consequently replace P by P—p(@,+6;—0,)/2n, Q by Q- p(O;+6,— G.)/2n, R by R— p(9, + O,— 63) | 2n, S by S+ pyr/n, T by T + pho/n, U by U+ prrs/n. With these alterations the equations (2) still hold good. In forming our criteria, we equate values found from (2) to values given by the stress-strain relations deduced from (3). In these latter P, Q, R, 8, T, U are to have the original values of these quantities and not those which replace them as above. Consequently, as our first step in the criteria, in place of V?r,=0 ete. we obtain nV] Nr, = evi, nV bro = pro, nV/ rs = pws, and similarly mx = PX 2V?X2 = PX» MV?Xs = px — (8) In place of the last stage which gave the metric criterion, we find Rui Th) 0°; , 0°: , 0°0: 3 (81 + 82+ 8s) — VA, +0, +95) +} Seat gat Gat snip OLX, ok Cee eal Se Once ean. 070; 20 ee 20) Ox: O'xe) 45 0°%3 a 2 x2, I°X3 : ei Oy Oz 22422) — ee and finally, po— —(mtn)V2h4+ 2m {54+ 2 Ox: ae us Os Yel ae a These are the modified form of (5) and (7). 3. Inclusion of bodily forces, with particular reference to gravity on the surface of. the Earth, and to centrifugal forces. In any case we shall have to consider the alteration made in equations (1), and their solution in (2) by the introduction 282. Mr. R. F. Gwyther on an Analytical Discrimination on the left-hand side of (1) of terms representing the com- ponents of the force per unit volume. These components can always be represented in a form similar to that given for the displacement in (3), but I shall suppose that the force per unit volume can be represented by the simpler forms pe TOE st Si) O(E +f.) OCH +73) OE uae ae ce os and I shail suppose that we have selected the axes and that the W’s are null. Then in (2) we must replace P by P+p(F +f), Q by Q+p(E +s), R by R+ pf tp) leaving 8S, T, U unchanged. In the melee found from the stress-strain relations there are no such changes to be made. We shall thus obtain from (6) VO: phi ora V0: — pfs = V?03— pj's 2n{3pF + p( fitsfo+fs)} + (3m +n)V?(O1+ Os + 4s) 0°0, . O'b7 OU: = 6m} =i at ye ime i, te) and Tf we ee rt ea = ++ +xX25 5 oe where $, ¥1, Xa, X3 the values in (5) and (7) and may be regarded as Complementary Functions, then we remain with V?°xX1' =a phis We: ae pi 2s V 2x3" == p73 2npk + (m+n)p(fitfetfs) + (m+nV7¢' 2 = 6m(2% 48 oe ox (11) which may be regarded as giving ‘s Soda Integral corresponding to the particular force acting. There are not many cases of interest. In the case of gravity on the surface of the Earth, as under natural forces generally, we have A=hr=p=0 and (m+n)V?d' + 2npk=0. If we suppose (—A, —p, —v) to be the direction-cosines of the attraction of gravitation, BP = —g(\a+ wy + vz) n Nee ee n) Ca! and and of Elastie Stresses in an Isotropic Body. 283 and if we write P’, Q’, R’ for the Particular Integral portion of P, Q, R—7.e. the terms which depend explicitly on g— we find m—n — gp(mytve), . « . (12) P’= gpxr JP tig a with similar values for Q' and R’, the Complementary Function part of P, Q, and R and the values of 8, T, U being those given in the earlier part of this paper. The other case which I propose to consider is that of a body moving with angular velocities @;, @,, @, about the axes of coordinates which must be axes fixed in the body. It is implied either that the question is purely kinematical, or that a problem in Rigid Dynamics has been previousiy solved. The expressions for the acceleration of a point in the body are well known, and give for the effect of the reversed effective forces F= Ho +02)0? + (@2+ o2)y'+(o2 +o) — —20,0,Ly — 20,0202 —20,0,y2}, with fi=2(yo:—2@,), fo=y(2@r—2oz), fs=clvay—yoz). The form of the forces f, /,, fs indicates that they will cause no strain in the body, and consequently cause no stress. If we proceed to find the effect which they have on the values of the stresses, they will be seen to disappear from the stress-equations. I shall therefore omit them for this purpose, and treat yy’, x2’, x3 as null. We then find n g = Ce {(@,? =5 ais" ya" + (@,” =f w.”)y* + (,? a3 w,")z* —Leyinyely?+2!)—2a,0.00(@ +2) —2owyxy(u? +y")} and pie pen Omn Oe) en A-(Sptga)? Seep at i Pe ke. (1B Ss war (13) with Complementary Function terms as before. These give no solution of any specific question. They only give a skeleton ot the general form which a solution will take. XXI. Note on Damped Vibrations. By H. 8. Rowe *. T is well known that the space time curve for free un- damped vibrations may be derived from the projection of a rotating vector, the end of which describes a circle, and it is fairly well known that for vibrations which are resisted by fluid friction proportional to the velocity, the space time curve may be projected (as remarked by P. G. Tait) from a rotating vector, the end of which describes an equi- augular or logarithmic spiral. The vibration of bodies when roseedl by a oonsteral frictional force-—say solid friction—is of great importance in practical work and does not appear to have been adequately treated. The results obtainable are, moreover, in themselves of much interest. If F is the constant force of friction the equation of motion is ma+c?e+F=0, wherein the sign of F depends on the direction of motion. Substituting z= X + F/c’, we have ae a= cos ( e == 55 e)) /m which gives a series of harmonic vibrations about alternating centres distant F/c? from the equilibrium position when ~* Communicated by the Author. ag Notices respecting New Books. 285 friction is absent. The motion can be obtained by pro- jection from a spiral which is composed of semicircles as shown in the diagram. It will be seen that the amplitudes are in arithmetical progression and the difference fur a complete period is 4I'/c’, which may be called the arithmetic decrement. There is little purpose in using the ordinary definition of decrement, but it may be remarked that on this definition (2. e. ratio of successive amplitudes) the decrement ranges from unity for infinite amplitudes to infinity for zero amplitudes. The spiral curve described here does not appear to have been used before in scientific work, and it might be con- veniently called the arithmetic spiral or the spiral of semi- circles. XXIT. Notices respecting New Books. Weather Prediction by Numerical Process. By Luwis F. Ricuarp- son, B.A., F.Inst.P. 4to, pp. xii+236. Cambridge Univer- sity Press. 30s. net. = usual method employed in weather forecasting is a development of that of Abercromby. Distributions of pressure are classified aceording to standard types, and the vari- ation on any occasion is predicted according to the behaviour of the atmosphere on previous occasions when conditions of the same type occurred. The method is therefore one of sampling inference in which the information utilized is all of one kind. Mr. Richardson believes that other information is relevant to the behaviour of the atmosphere; and in this book he shows how to make use of the known results expressed in the hydrodynamical equations of motion and the equations of emission, transference, and absorption of heat and water. The method adopted is to work with equations each containing only one partial differential coefficient with regard to the time, so that this can be determined by means of the equation when the other quantities involved are known; they include, of course, partial derivatives with regard to the position on the map and the height. These are to be found by observation at stations distributed according to a regular pattern, and the rate of change of each meteorological element at each station is to be calculated from them. Complications arise from the facts that the observations must be made at finite intervals both of time and of position, but these are allowed for. The stations required are more numerous than those at present in operation, and observations should be made every three hours to obtain the best results. Observations of upper-air Winds and temperatures are required. 286 Geological Society :— The method is one that appeals strongly to the mathematical physicist. It is necessarily laborious in its present form, and probably could not be worked with sufficient speed to make it a practical method of forecasting; but when forecasters have acquired experience in its use, they will probably find that a sufficient number of the quantities allowed for are comparatively small to make it possible to expedite the calculation considerably without great sacrifice of accuracy. The value of the work is not confined to the application to forecasting, though the possibility of predicting the disturbing occasions when cyclones cause merriment in the daily press by moving in the wrong direction makes this the feature of most general interest. Its discussion of the physical properties of the atmosphere is so thorough that it constitutes a text-book of the subject. Copious references to original literature are piven, and - any meteorologist requiring serious information on any topic will do well to look first in this book. The section on evaporation suggests that the only limitation on the evaporation from vege- tation is imposed bv the dithculty of passing along the stomata tubes; this is not always true even for an isolated leaf, and is certainly wrong for a carpet of grass, on account of the obstruc- tion offered by the vapour from one stoma to evaporation from another. The numerical data actually given, however, eliminate this source of error. Concerning the printing and style of the book, it is only necessary to say that it is published by the Cambridge University Press. The index is good. Hd XXUI. Proceedings of Learned Societies. GEOLOGICAL SOCIETY. [Continued from vol. xlili. p. 1186.] February 1st, 1922.—Mr. R. D. Oldham, F.R.S., President, in the Chair. Mr. Cyrit Enywarp Nowrit BromeneaD, B.A., F.G.S., de- livered a leeture on the Influence of Geolog sy (On “nae History of London. The 6-inch Geological Survey maps constructed by the Lecturer were exhibited, and some of the new features pointed out. The small streams now ‘buried’ are indicated on the maps, and the historical research involved in tracing them led to an appreciation of the connexion between the geology and topography on the one hand, and the original settlement and gradual growth of London on the other. The reasons for the first selection of the site have been dealt with by several writers: below London the wide allu- vial marshes formed an impassable obstacle; traffic from the Influence of Geology on the History of London. 287 Continent came by the ports of Kent, and, if destined for the north or east of Britain, sought the lowest possible crossing of the Thames. This was near old London Bridge, where the low- level gravel on the south and the Middle Terrace deposits on the north approached close to the river-bank. A settlement was obviously required here, and the northern side was chosen as the higher ground. The gravels provided a dry healthy soil and an easily accessible water-supply; they crowned twin hills separated by the deep valley of the Walbrook, bounded on the east by the low ground near the Tower and the Lea with its marshes, and on the west by the steep descent to the Fleet; the site was, therefore, easily defensible. ‘The river-face of the hills was naturally more abrupt than it is now, owing to the reclamation of ground from the river; the most ancient embankment lay 60 feet north of the northern side of Thames Street. The first definite evidence ef a permanent settlement was the reference in Tacitus. The early Roman encampment lay east of the Walbrook, and the brickearth on the west around St. Paul’s was worked. Later the city expanded, until the St. Paul’s hill was included, the wall being built in the second half of the 4th century. The great Roman road from Kent (Watling Street) aveided London, and utilized the next ford upstream—at West- minster—on its way to Verulamium and the north-west. The earliest Westminster was a Roman settlement beside the ford, built on a small island of gravel and sand between two mouths of the Tyburn. This settlement could not grow, as did London, since the area of the island, known to the Saxons as Thorney, was small. The road from London to the west joined the St. Alban’s road at Hyde Park Corner, running along the ‘ Strand,’ where the gravel came close to the river; a spring thrown out from this gravel by the London Clay was utilized for the Roman Bath in Strand Lane. Throughout Medizval times London was practically confined to the walled city, a defensible position being essential. The forests of the London-Clay belt on the north are indicated in Domesday Book and referred to by several writers, notably Fitzstephen, whose Chronicle also mentions many of the springs and wells and the marsh of Moorfields, produced largely by the damming of the Walbrook by the Wall. The same writer mentions that London and Westminster are ‘connected by a suburb.’ This was along the ‘Strand,’ and consisted first of great noblemen’s houses facing the river and a row of cottages along the north side of the road; this link grew northwards, at first slowly, but in the second half of the 17th century with great rapidity. By the end of that period the whole of the area covered by the Middle-Terrace Gravel was built over, but the northern margin of the gravel was also that of the town for 100 years, the London- Clay belt remaining unoccupied. The reason for this arrested development was that the gravel 288 Intelligence and Miscellaneous Articles. provided the water-supply. In early days the City was dependent on many wells sunk through the gravel, some of which were famous, such as Clerkenwell, Holywell, and St. Clement’s. In the same way the outlying hamlets (for instance, Putney, Roehampton, Clapham, Brixton, Haling, Acton, Paddington, Kensington, Islington, etc.) started on the gravel, but later outgrew it, as pointed out by Prestwich in his Presidential Address of 1878. In the City the supply soon became inadequate, or as Stow says ‘decayed,’ and sundry means were adopted to supplement it. The conduit system, bringing water in pipes from distant springs, began in 1236; London-Bridge Waterworks pumped water from the Thames by water-wheels from 1582 to 1817; the New River was constructed in 16138, and is still in use. It was not until the 19th century that steam-pumps and iron pipes made it possible for the clay area to be occupied, thus linking together the various hamlets that are now the Metropolitan Boroughs. Some of the ways in which Geology affects London to-day were briefly indicated, and the lecture was illustrated by a number of lantern-slides, reproduced mainly from old maps and prints. XXIV. Intelligence and Miscellaneous Articles. YOUNG'S MODULUS AND POISSON’S RATIO FOR SPRUCE. To the Kditors of the Philosophical Magazine. Dear Sirs,— (A my recent paper in the Philosophical Magazine for May 1922 there is an error on page 877. It is there stated that Sve Ser es Se eae gel eae ee hee ADoser S are equal respectively to 7008,, 700 8, and 5: This should read are equal respectively to 700 S,, 700 8, and 3: The error becomes evident on reading the paper, but I very much regret that it has crept in. Yours faithfully, The College of Technology, H. Carrineron, Manchester. May 24th, 1922. Tizanp & Pyn. lanGele Corresponding to experiment A,.. iG, B. Corresponding to experiment A,,. In these photographs the lower horizontal line is the line of atmospheric pressure. The ordinates represent pressure, and the abscisse time. A is the beginning of compression, B the point of maximum compression, and C the explosion. The curve in the top left-hand corner is the cooling curve of the products of combustion. Phil. Mag. Ser. 6, Vol, 44, PI. Corresponding to experiment A,,. ine Corresponding to experiment Ds. [Qnree ye Corresponding to experiment Ds. THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCE ASO [SIXTH SERI AGPGT ST A922: XXV. On the Viscosity and Molecular Dimensions of Gaseous Carbon Oaysulphide (COS). By C. J. Smirz, B.Sce., A.R.CS., DIC., Research Student, Imperial College of Science and Technology™. | HE present research is a continuation of the work on the measurements of the viscosities of gases, for the purpose of elucidating the structure of the molecules constituting them. Some measure of success has attended this investigation in many cases where the necessary data are known, and suggests that an accumulation of further similar data may be fruitful. A case in point js that of the molecule of carbon oxysulphide, and this paper describes the measurements of the viscous properties of this substance, which is ordinarily gaseous. The data, hitherto unknown, which have been obtained, have been applied to calculate the molecular dimensions in the ordinary way. Apparatus and Method of Observation. The apparatus and method, which have been used to determine the viscosity of carbon oxysulphide, have recently been fully described Tf. : Method of Haperiment. The viscometer was carefully standardized with a new mercury pellet in the manner indicated in previous papers. * Communicated by Prof. A. O. Rankine, D.Sc. + A. QO. Rankine and C. J. Smith, Phil. Mag. vol. xlii. p. 601, Nov. 1921, and C. J. Smith, Proc. Phys. Soc. vol. xxxiv. p. 155, June 1922. Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. U 290 Mr. C.J. Smith on the Viscosity and Molecular and the corrected time of fall proved to be 104-70 sees., a value which is probably correct to 0O'1 sec. With this time of fall the corresponding time of fall for carbon oxysulphide has been compared, and with appropriate corrections gives the relative viscosity, from which the absolute viscosity has been obtained by assuming the viscosity of air at 15°-0 C. to be 1°799 x 1074 C.G.S. units. In addition, the variation of viscosity with temperature has been derived from com- parisons of the corrected times of fall at atmospheric and steam temperature. Preparation and Purification of the Carbon Oxysulphide. The carbon oxysulphide was prepared by the action of sulphuric acid (five vols. acid, four vols. water) on pure potassium thiocyanate in the cold (room temperature). At the same time hydrocyanic acid, formic acid, and carbon bisulphide are formed. To remove these impurities the method recommended by Moissan* was used. ‘This consists in passing the gas through a strong solution of caustic potash to remove the hydrogen cyanide and then over wood charcoal to remove the carbon: bisulphide. The gas was dried by being passed over calcium chloride, and then solidified at liquid air temperature. All permanent gases were pumped out of the U-tube containing the solid COS by means of a mercury pump. The liquid air was then replaced by a mixture of solid CO, and alcohol at —80° C., when it was observed that the vapour pressure of the liquid COS was about 30 cm, of mercury. The CO, mixture was then removed and samples of the gas collected over mercury. It was further purified before being introduced into the viscometer by fractional distillation at liquid air temperature. The liquid air having been replaced by CO, and alcohol at —80° ©., it was possible on account of the comparatively high vapour pressure of COS at this temperature to pump off successive quantities of dry COS sufficient to fill the viscometer at atmospheric pressure. Heperimental Results. (Table I.) We have ¢,;;=69°96 sees., and ty)=90'64 sees. The ratio of these times of fall gives the ratio of the viscosities at the corresponding temperatures. Thus moo = too _ 90°64 _ 1 5x6 pis ate OOOO) GF * Moissan, Zraeté de Chime, vol. 1. p. 318, i” P Dimensions of Gaseous Carbon Ovysulphide. 291 TABLE I. Each time recorded is the mean of four observations in each direction for the whole pellet, and of three for the pellet when divided into two segments. The letters in parentheses indicate the order in which the observations were made. ma : ee of ay (segs. ). | Capillary Corrected | Time Ate (deg. C.). | Whole | Two | Ae a 15°00. 100°-0C. | pellet. | segments. | | | | (a) ipag...| 70L | 7685 | 00416 | <7001 | 6992) — (0) 15-44... | 7308 | 7625 | 00399 | 70-16 | 7005) — (e) 15°68 ... 73-02 76°22 0:0403 | 70-07 | 69°91 | — | | Mean | 69:96) — forage ce. |, OT51 92°43 | 0:0099 | 90°60 -— 90°63 a) deo...) | 91-54 9248 | 00101 | 9062 — 90°65 Mean — 90°64 Assuming Sutherland’s law to hold for this gas, the value of Sutherland’s constant obtained is C=330. miso at 15-0 C., tcos id 69°96 fe ATT) Correcting for difference of slipping of COS and air, we obtain =O 0G 2. JESS Ug ane lair On the assumption that the viscosity of air at 15°-0 C. is 1:729 x 1074 C.G.S. units, the values for COS are 7 1200 x10E* CGS funits and Nig = L504 x 107* C.G.S. units: and by extrapolation using Sutherland’s formula, n= lel 3a x10 C.G. Saamits: Calculation of Molecular Dimensions. The particular dimension calculated from the above results is the mean area A which the molecule presents in mutual collision with others. The basis of this calculation is Chapman’s formula (doc. cit.) modified in its interpretation in the manner suggested by Rankine. The value obtained is A=1:06 x 10~-” cm.? which may be subject to an error of 2 or 3 per cent. U2 292 -Prof. A. O. Rankine on the Molecular Structure Summary of Lesults. tein als Viscosity in C.G.S. units x 10-4. | | -Sutherland’s 1 Mean “| constant. |°° ene 15°:0 C. 160°:0 C. 0°:0 CG. cm. X i 1-200 1°554 TeI35)) on 1-06 Tn bdnein ee the author gladly acknowledges the grant for this research, which was made by the ore ee Grant Committee of the Royal Society, and also wishes to thank Professor Rankine for his continued help and advice. Imperial College of Science and Technology, S.W. 7. Ist May, 1922. XXXVI. On the Molecular Structure of Carbon Oxysulphide and Carbon Bisulphide. By A. O. Ranking, D.Se., Professor of Physics in the Imperial College of Science and Technology *. Ie HERE are at the present day in the process of development several theories of atomic and mole- cular structure which are in many respects discordant. They have, however, at least one feature of general agree- ment namely, the common view that the aieme of the inert gases occupy unique positions in the various schemes. The distribution of the electrons with respect to the nuclei in these atoms is regarded as having the characteristic of completeness, so that there is displayed no marked tendency to lose electrons or to capture additional ones. Moreover, atoms other than those mentioned are believed to have in varying degrees what may be called deficiencies and redundancies of extra-nuclear electrons, which they endeavour to adjust by entering into suitable combinations with one another ; so that either by the process of give and take, or by common use of the same electrons, contigur ations corre- sponding closely to those of the inert atoms are attained by the individual atoms forming the compound. 2. These views of chemical combination find site most * Communicated by the Author. of Carbon Oxysulphide and Carbon Bisulphide. 293 complete expression in the theory of Lewis and Langmuir *, particularly in relation to the type of compound with which this paper is concerned—namely, that in which atoms, deficient in electrons, are regarded as sharing them in euler to reach the completeness of inert configurations. The main purpose of this paper is to apply the principles of this theory to the special case of the molecule of carbon oxysulphide, and to show that the molecular dimensions of this compound, as derived from viscosity data, are consistent with the Lewis- Langmuir view of its constitution. This test of the validity of the theory is made possible by the recent measurements by ©. J. Smith t of the viscous properties of the gas in question. Similar calculations for the molecule of carbon bisulphide have been made, and these await verification or otherwise when the necessary viscosity data are availabie. 3. Carbon oxysulphide belongs to a family of three ola having the chemical constitutions GO, COS, and CS,. The two former are gaseous at ordinary temperatures, and the latter a highly volatile liquid. Inall of them carbon is a constituent, and COS can he regarded as the molecule obtained by the substitution of a sulphur atom for one of the oxygen atoms in COs, or by the reverse substitution in CS,. It is probable that the carbon atom occupies the central position in each molecule, and that the nuclei of the three atoms lie in each case upon a straight line. 4. According to the Lewis-Langmuir theory (loc. cit.), the atoms in these molecules are linked together by sharing external electrons in such a manner that each atom approxi- mates to the configuration of the inertatom at the end of the corresponding row in the periodic table. Thus, in CO, the ceutral carbon atom shares altogether eight electrons, four on each side with an oxygen atom. The electron configuration thus formed is that of three neon atoms in a row, for the inert atom corresponding to both carbon and oxygen is neon. In the molecule COS there are again eight electrons shared by the carbon atom, four on one side with the oxygen atom, and four on the other side with the sulphur atom. The electron arrangement thus attained is that of two neon atoms (corresponding to the oxygen and carbon) and one argon atom (corresponding to the sulphur). Applying a similar argument to the CS, molecule, we are led to regard it as resembling closely the electron distribution of inert atoms in the sequence argon-neon-argon ‘in a line. In other words, we can treat each carbon or oxygen atom in a * IT. Langmuir, Journ. Amer. Chem. Soe. vol. xli. p. 868. + C.J. Smith, supra, p. 289. 294 Prof. A. O. Rankine en the Molecular Structure molecule as having nearly the same dimensions as a neon atom, and each sulphur atom in combination as approxi- mating to the dimensions of an atom of argon. 5. The remaining question of how far apart are the nuclei of the atoms in the molecule finds a satisfactory answer in the work of W. L. Bragg *, whose X-ray -crystal measure- ments have enabled him to assign probable values for the radii of the-outer electron shells of the atoms of the inert gases. The only values with which we are at the moment concerned are those of neon, and argon, which are given respectively as 0°65 and 1:03 Angstré jm units. In cases like those under consideration, where outer electrons are playing a double part, the sharing is equivalent to contiguity of the outer shells, so that the distance apart of the nuclei is the sum of the radii of the appropriate inert atom shells. Thus for CO,, which is pictured as three neon atoms in line, the three nuclei are equally spaced and separated by distances equal to twice the radius of the neon outer “shell, 7. e. -2x0:65 A=—1:30 A. In COS the distance between the carbon and oxygen nuclei is the same, namely 1°30 A, but the distance between the carbon and sulphur nuclei is the sum of the radii of the outer, electron, shells of the neon and argon atoms, 2. e. 0°65 A+1:03 A=1°68 A. The three nuclei in COS are thus unequally spaced on _ account of the greater size of the argon atom. In CS, the distance between the, carbon nucleus and each sulphur nucleus is also 1°68 A, and the three | nuclei are again spaced symmetrically. 6. It is evident that none of the three molecules under consideration, if their configurations are as indicated, can be expected to display spherical symmetry. In these circum- stances it is necessary to interpret in a special way the results of the well recognized method of caleulating molecular dimensions from viscosity data. The quantity which is actually derivable from the formula is the mean value of the area which the molecule presents, for all possible orientations, as a target for mutual collision with other molecules in the gas. This area the author + has ventured to call the mean collision area, and its value for COS is given by C.J. Smith (loc. cté.) as 1°06 x 107 em.? ‘The immediate problem before us is to find how nearly the tentative model of this molecule described above would exhibit this value for its mean collision area. The values of * W. L. Brage, Phil. Mag. vol. xl. p. 169. 7 AgeO, Rankine, Proe. Roy. Soc. A, vol. Sead p. 360, and Proc. Phys. Soc. vol, xxxili. p. 362. of Carbon Oxysulphide and Carbon Bisulphide. 295 the mean collision areas of the constituent configurations (which we are taking to be those of neon and argon) are known, and it is usual to. regard these symmetrical inert atoms as behaving as elastic spheres for purposes of collision. The radii of these collision spheres, as we may: call them for the sake of precision, are 1°15 A and 1:44 A respectively, and they are considerably larger than those of the corre- sponding outer electron shells, so that they overlap when Fig. 1.—Molecular Dimensions from the point of view of the Kinetic Theory. os The Carbon Dioxide Molecule: equivalent to three linked atoms of Neon. ena § The Carbon Oxysulphide Molecule: equivalent to two Neon atonis and one Argon atom linked together. The Carbon Bisulphide Molecule: equivalent to two Argon atoms linked together by one Neon atom, the nuclei are separated by the distances demanded by electron sharing. Fig. 1 shows three models, drawn to scale, representing what we may conceive CO,, COS, and OS, to be like for purposes of intermolecular encounters. CO, may be regarded as three overlapping spheres, each of the neon 296 Prof. A. O. Rankine on. the WMoleould® Seruature collision size, with centres separated by the distances already specified. In COS we take instead of one of the extreme neon spheres an argon collision sphere ; while in CS, both the outer spheres are of the argon size. In all three cases ithe diagram represents all the nuclei in the plane of the paper, and the line joining them is evidently an axis of symmetry. If these symmetrical axes are variously oriented, the area presented by the model assumes different values, and our problem is to caleulate the mean value of this pro- jected area for comparison with that deduced from viscosity data. The author (loc. cit.) has already derived the necessary formule for this purpose, and has shown that the result obtained by application to the first model in fig. 1, namely CO., is very nearly equal to the actual mean collision area of the carbon dioxide molecule. In other words,. a carbon dioxide molecule behaves in collision as though it had the configuration of three neon atoms in a straight line and with outer electron shells contiguous. Calculation for the COS Model.—In the model which we are taking to represent the COS molecule, the calculation in the strictest sense is greatly complicated by reason of the particular distribution of the spheres. The exact formule which have been obtained (loc. cit.) for equal and unequal spheres only apply rigidly to cases where a special relation exists between the diameters of the spheres and the distances apart of their centres; and the model under consideration does not fulfil this condition. But by regarding the problem from two different points of view, we can obtain, by means of the comparatively simple formule: already availakle, upper and lower limits which are so close together as ic render unnecessary the laborious exact calculation. This course is all the more justifiable because it is fully recognized that the general treatment of the problem itself can only be taken as a first approximation to the truth. 8. Let us consider the effect on the area of projection of the model (reproduced in the full lines of fig. 2, a) caused by variations of orientation of the symmetrical axis joining the centres O,, O., and O; of the constituent spheres. It will be convenient to speak of the sphere with centre O, simply as sphere 1, and so on, and of the projections of the spheres, which will of course be circles, as projection 1 etc. «As the axis O, O3 approaches the line of sight, the projections of the centres approach one another, and the eclipsing of the spheres becomes more and more marked. Up to a certain point the total projected area is equal to the sum of Carbon Oxysulphide and Carion Bisulphide. 297 of the areas of the whole of projection 3, the crescent formed by the overlapping of projection 3 over projection 2, and the crescent formed similarly by the eclipse of projection 1 by projection 2. Before the eclipsing of 2 by 3 is complete, however, projection 3 begins to encroach upon regions of projection 1 which are not already covered by projection 2. It is this fact that introduces into the exact treatment of the problem the complications to which reference has already been made. Thus in fig. 2,6, which shows the projected area for that orientation of the axis for which the eclipse of 2 by 3 is just complete, the crescent formed by 2 and 1 still -4- mest oes ag fs -}+---4-- i / / 0, 03 0; / survives, but parts of it (as indicated by the shading) are covered by projection 3. The projected centres are O,’, O,’, and QO,’ respectively, and this particular state of affairs occurs when the angle between O, O; and the direction of projection is 9° 47’ for the spheres having the dimensions and distribu- tion already specified. 9. Overlapping of the type just indicated, like all overlapping, has the effect of reducing the projected area ; it is therefore clear that if we neglect it we shall obtain too large a value for the mean area of projection—that is, an upper limit will be obtained by taking the mean collision area as the sum of the three parts : (a) the area of the circle 3, (b) the mean value of the area of the crescent formed by ites 3 and 2 (c)the mean value of the area of the crescent 298 Prof. A. O. Rankine on the Molecular Structure . formed by circles 2 and 1. The first of these quantities is. the area of the central cross-section of the argon sphere itselr, viz. 0°648x107-" cm.?; the two latter are readily obtained from the graph in the paper already mentioned * They prove to be 0-217 x 10° ™ em? and 0;226> 109 Senn respectively. The total is 1:09 x 10~” em.’, and this provides. our upper limit. 10. With regard to the lower limit, we can obtain a. satisfactory value by contemplating a variation of our model, which avoids the special type of overlapping responsible for complications. A suitable change for this purpose is to substitute for the sphere 1 a smaller sphere having the same centre but of such magnitude that its projection becomes just eclipsed by projection 2 at the same orientation of the symmetrical axis for which projection 2 is just eclipsed by projection 1, as shown by the dotted circles in fig. 2. The radius of the necessary sphere is found to be 0:93 A as compared with the original value 1:15 A. [Examination of the projection of a sphere of this size, in relation to the other: two projections, shows that for no orientation does eclipsing of the shaded type appear, and the formule already available enable the mean area of projection to be calculated exactly. The value so obtained will, however, obviously be less than the true value aimed at, on account of the reduction of size: assumed for sphere 1. Using the graph already mentioned, the lower limit thus derived is 0°648 x 105” cm.? 4:0°217 x 10> © em? -+.0°138 x 105 mire = 1°00 x 10 get iemme 11. The foregoing justifies the assertion that a molecular model having the dimensions of an argon atom succeeded by two neon atoms in line and spaced according to the demands of outer electron contiguity may be expected to. have a mean collision area intermediate between J)S) 5 1 ean. and LeOO se O72 oin2 The actual value of the mean collision area of the COS molecule, as determined from viscosity is DO << WO ena with a possible error of 2 or 3 per cent. It falls definitely between the upper and lower limits obtained from our calculations, and seems to provide striking corroboration of 7 XO vankine, Proc wehys, Soc; vol, aacxai. app ole of Carbon Oxysulphide and Carbon Bisulphide. 299 the theory upon which the estimates are based. But we must be content with the conservative remark that the dimensions of the carbon oxysulphide molecule, as found by the application of the kinetic theory, are consistent within the limits of experimental accuracy with the view that the three atoms of the molecule, by sharing external electrons, assume the electron configurations and ee wounk in collision of particular groupings of the neighbouring inert atoms. 12. Calculation for the CS, Model. —Although there exist at present no data for carbon bisulphide which suffice to calculate the mean collision area of the molecule in the gaseous state, the success of the previous comparison would appear to justify a prediction of its value by con- sideration of the appropriate model. This has been repro- duced in the full lines of fig. 3,a. Here again the mode! is Fig. 3. (2) one which does not lend itself to exact solution without laborious calculation; but again, also, we can obtain satisfactorily close upper and lower limits. The area of projection will clearly be less than that corresponding to the mode! in which the dotted sphere is substituted for the small central one, so that we have three equal spheres of the argon size in line ; ; it will, on the other hand, be greater than if the central sphere i is entirely dispensed with , so that ae are two equal argon spheres only, as represented in fic. 3,b. The dimensions of the spheres and the distances 300 Mr. Hf. P. Slater on the: Ruse of apart of their centres have already been stated ; and both modified models have mean areas of projection which are very easily calculated. The upper limit thus determined proves to be 2°12 times the collision area of the argon atom ; the lower limit is 1°90 times the same area. Using the known value 0°648 x 107)? em.? for. the collision area of the argon atom, we find that the mean area of projection of the model consisting of two argon atoms with an intermediate neon hes between 13 Creeks and 1223 Al? ema We may venture to predict with some confidence that the mean collision area of the CS, molecule, when determined, will be found to be between the above vaiues. A more exact estimate could of course be made, but the degree of accuracy at present attainable in determining molecular dimensions from viscosity measurements is not sufficient to render the additional calculation worth while. Summary. On the assumption of the validity of the Lewis-Langmuir view of molecular constitution, the probable betaviour during encounters has been examined for the molecules of carbon oxysulphide and carbon bisulphide. In the former case it is shown that the molecular dimensions as derived from the application of the kinetic theory to the viscosity measure- ments of C. J. Smith, are in striking accordance with the results of the above examination. In the latter case comparison is not yet possible, on account of the absence of necessary data. Imperial College of Science and Technology, May 11th, 1922. XXVIL. The Rise of y-Ray Activity of Radium Emanation. iy. P. SLATER WUE SCay ica aso. Canmaue ae N a previous paper f it has been shown how the initial rise of y-ray activity, starting from pure radium emanation, depended on the nature of the walls of the tube containing the gas, the reason .being that a small but * Communicated by Prof. Sir E. Rutherford, F.R.S. + Slater, Phil. Mae. vol. xi. p. 904 (1921). y-Ray Activity of Radium Emanation. 301 detectable y-radiation was excited in the walls by the impact of the & particles emitted by the emanation. The amount of this excited radiation was, however, very small when the walls of the tube were composed of atoms of low atomic weight, and for a lining of pure paper the y-ray aetivity of the emanation and its products was found to rise practically from zero. Under such conditions the y radiations from the tube are due only to the products radium B and radium C. Taking the number of emanation atoms disintegrating per second at initial time as unity, the number of radium-B atoms disintegrating per second at any subsequent time ¢ is Aor >a (Ay—Ay)(Ag—Ay)’ X=1, 233 where A,, Ay, As are the transformation constants of the emanation and the products A, B, and © aay aN This quantity is tabulated for various times up to 220 minutes at the end of this paper (Table II.). Similarly, the number of radium-C atoms disintegrating per second at time ¢ is e-Ale eg: 2, (Ag — Aq) (A3—Ay} (Ag Ay)’ A= 1, 2,3, 4 Tables for this quantity for various times up to 258 minutes have been given by Moseley and Makower * and by Rutherford f. The rise in y-ray activity of a tube filled initially with pure emanation can therefore be represented by e-Aié KA.A3> (A2— Ay) (As — r1) N=1,2,3 e Ait Su meee (ga) s— WJM) where K is the fraction of the ionization, measured under given absorption conditions, due to radium B when in radio- active equilibrium with radium C. Thus it is necessary to determine “K.” Since the y rays * Moseley and Makower, Phil. Mag. vol. Xxili. p. 302 (1912). 1 Rutherford, ‘R adioactive Substances s, p. 499. 302 Mr. F. P. Slater on the Rise of from radium B are less penetrating than those from radium C, “ K” depends on the thickness of matter through which the radiations pass before entering the ionization chamber. Rise curves have been experimentally determined for different thicknesses of absorption material, both lead and aluminium being used. The values of K for various thick- nesses have been deduced by trial, and are shown in fig. 1. ie 15 20 —-~ Moms. c fead. O i) A comparison of the experimental and calculated rise curves of the y-ray activity through 12°0 mm. of lead is given in fig. 2, After six minutes from the introduction of pure emanation, the calculated and experimental curves agree very closely. From these curves the absorption coefficient of the radium B-y rays can be deduced, and the values found are given in Table I. along with comparative determinations by Makower and Moseley (loc. cit.) and Rutherford and Richardson * The values of the absorption coefficients for the thick- nesses of aluminium are somewhat doubtful, since the supposition of homogeneity of the radium-C y rays is not justifiable through such small thicknesses. The increasing * Rutherford and Richardson, Phil. Mag. vol. xxv. p. 722 (1913). y-Ray Activity of Radium Emanation. 303 value of w (cm.~!) with decreasing thickness of absorption material (see Table I.) is to be expected, since Rutherford and Richardson (loc. cit.) showed that radium B emits certainly two types of radiation having absorption coefficients in aluminium of 0°51 em.~! and 40:0 em.~’, and possibly a third type (w= 230°07" in aluminium). Fig, 2.—Rise of y activity from Radium emanation through 12:0 mm. of lead. a % Max. Activity \ 0-60 0-50 0-40 0:30 0:2 0-10 The absorption coefficients in lead only, given in Table I., are corrected for obliquity of the rays entering the electro- scope, and King’ s correction is used as given in Case II. of his paper * _f (wt)—cosOf[mtsec@] t= 1—cos 0 where I; and I, are the intensities of the radiation emerging through a plate of thickness ¢ cm. and incident radian respectiv ely, w the absorption coefficient expressed in cm.™’, and @ the semi- -angle of the cone of rays entering the electroscope. * King, Phil. Mag. vol. xxiii, p. 248 (1912). 304 Rise of y-Ray Activity of Radium Emanation. TARE aeY 3) Absorbing medium is Lead, except where otherwise shown. Value of p (cm.—1) Moseley Thickness of Rutherford Absorbing Plate. ae ate one e Radium-B rays. Makower. Richardson. 1G O26: Ohm mis eis. 2. ees. 2) 27 tem) 1 — Varying from TOO SSO. Oe oe A ew Ga a. 11-0 cm.—1 QE ROS) pc” empl 3 sbss a Brars 4-1 em,-1 4:0 em.-1 to (lead). 2°8 cm.—1 in ISS Se. 2 hol aan 62 cm.—1 6:0 em.—1 lead. (lead), 3°0- 4:0 ,, (Aluminium), i (ema CAlye mm. (Aluminium) TO:Otenn al (Al) & ae hie ele Rise of Radium B from Radium Emanation. Maximum = 0-97480 is taken as unity. Calculated Calculated Calculated Time ‘Time in rise of 1D rise of in ‘rise of mins Radium B. mins. Radium B. mins Radium B. 1 PEs Cn 0:00269 Vale ee 0:2260 TO Pee 0:9502 De EOC Nea 0-O1016 NG teste toeeee 0:2649 120) a oe 0:9643 Sie an 0:02129 lowe ae 03023 Ould eee 0:9750 Ais 1 aes 0:035138 Ie Oca ceehcte: 0:3379 PAG) no Ae ee 0°9835 Lee tae 0:05115 12 (SSK Nreeaitle e 0°4942 TSO a eee 0:9886 Gouna 0:06862 he SN 06185 160 eee 0:9929 "Tne 5 aoe 0:08718 A te 0:7066 Ree 0°9956 So. ee 0 10647 IN SC 6 Hautes 0'7780 [S02 ee 09979 GF, ee 126929 Moe ee 0:8329 19025 Rue 0:9992 LQ 2k eee 0'14624 SOie) alae 08748 ZOO eee 0:9998 emertee ols 0°16637 OU) eee anae 0:9058 DLO* a eecaaee 0:9999 Oy me 018641 TOO a ae 0:9315 Die eee 1:0000 DEA) aikeee ae 0:9999 Summary. Curves showing the rise of y-ray activity from pure radium emanation measured through a wide range of absorption thickness of matter have been determined and utilized in deducing the absorption coefficients of the hetero- geneous y¥ radiation from radium B. My thanks are due to Professor Sir KH. Rutherford for his invaluable help in carrying out this research, and to Mr. G. A. R. Crowe for the preparation of the radioactive material, Eo B0aa5q XXVIIT. An Experimental Test of Smoluchowski’s Theory of the Kinetics of the Process of Coagulation. By JNANENDRA Nara MouxuerJen, /).Sc., Professor of Physical Chemistry, University of Caleutta, and B. CoNSTANTINE PaAPacon- STANTINOU, D.Sc., Assiscant Professor of Chemistry, Uni- versity of Athens ™*. A short account of the Theory. aT some experiments on the degree of dispersion of colloidal arsenious sulphide on the rate of coagulation, it has been shown (J. Amer. Chem. Soc. vol. xxxvil. p. 2026, 1915; and Sen, Trans. Chem. Soc. vol. cxv. pp. 467-8, 1919) that the finer sol is less stable. In 1915 one of us pointed out the obvious connexion with the increased facilities of coalescence. The smaller particles have a more vigorous Brownian movement due to the smalier frictional resistance of the medium. ‘This would be clear from the well-known equation of Hinstein. The diminution in the mean distance between the particles also increases the rate of collisions. It was stated that the adsorption theory does not take these factors into consideration. Recently Smoluchowski (Zeit. Phys. Chem. vol. xcii. p- 129, 1917) has been able to formulate the progress ot the coalescence with time. His attention was drawn to the subject by Zsigmondy. Bredig (Anorganische Fermente, 1901, p. 15) suggested as the cause of coalescence an increase in surface tension with a decrease in the electric density on the particles. Zsigmondy (Zertsch. Physikal. Chem. vol. xcii. p. 500, 1918) medified this idea in the sense that there is an attraction, between the particles which increases with decrease in the electric charge. As a result of this attraction he assumes that when one particle coines within a certain distance of another, the two coalesce. This distance is taken as a measure of the force of attraction and is called the radius of the sphere of action. I[t has been shown by Zsigmondy that the time required for a definite colour-change in a gold sol gradually decreases with rise in electrolyte concentration till it reaches a minimum +, which does not change any further with higher * Communicated by Prof. F. G. Donnan, F.R.S. + Similar minimum times have been observed with cupric sulphide and mercuric sulphide sols by the writers. A copper sulphide sol gave two minutes as the time necessary for the appearance of visible clots when the concentration of the precipitating electrolyte (barium chloride) was varied from N/300 to N/20. At dilutions higher than N/300 the time was observed to increase as usual (Mukherjee and Sen, Joc. cit.). Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. X 306 Profs. J. N. Mukherjee and B. C. Papaconstantinou on concentrations. This was assumed to prove that the radius of attraction reached a maximum value. Smoluchowski utilized this idea of a sphere of action to avoid a consideration of the forces that influence the coalescence. He considers the probability of particles coming within their mutual sphere of action when the radius of the sphere has a constant value determined by the conditions.- It is assumed that as soon as a particle comes within the sphere of attraction by virtue of its Brownian movement the two particles coalesce. This dis- continuous view of the obviously continuous process of coalescence was assumed to avoid a consideration of the nature and distribution of the forces that are present. Considering the effect of the motion of each particle and also that each of the aggregates acts as a condensation centre, he derives the following equations : n=, (1) I+ ip y= ae (2) (145) ) (a, ngt)” eae w here “2.” denotes the total number of particles originally present per unit volume before coalescence begins. They are all assumed to be-spherical and equal in size. ‘“¢” is the time in seconds that has elapsed since the electrolyte and the sol have been mixed. “TT” is a constant charac- teristic of the rate of coagulation and is given by 1 Ta aa) Site nioke 1 ane eomlede where ‘‘D”’* is the diffusion constant as given by Hinstein’s ‘equation; #«=4.a7.D.Ra, and Ra is the radius of the sphere of uate. HH, ule ] * BD) MN SEE) te where H=the gas constant, 0 Hie 6=the absolute temperature, N)=Avogadro’s number, n=the viscosity, and =radius of the particle. the Kinetics of the Process of Coagulation. 307 yn denotes the total number of particles in all stages of coalescence in unit volume when the time is <4? ; ; n, denotes the number of the primary particles whose original number was 7 at the time “t” ; nz denotes the number of particles of the kth stage of coalescence—that is, the number of aggregates each of which consists of ‘ h” of the primary particles. ‘“‘k” is evidently an integer. In 1918 Zsigmondy published the results of an investi- gation to test this theory. He restricted his investigation to the rate of decrease in the primary particles (green in the ultramicroscope) in a colloidal gold sol when the minimum time of coagulation has been reached. He found that Ra=2°2 times r, the radius of the particles. Similar values were obtained by Westgren and Reitstétter (Zeitschr. Phys. Chem. vol. xcii. p. 600, 1918) with more coarsely dispersed gold sols. The value of Ra/r, however, varied in one experiment from 1-4 to 3°8. The recent experiments of Kruyt and Van Arkel (Rec. Trav. Chim. Pays-Bas, vol, sextrx, [4] p. 656, vol. xl. p. 169, 1920) show greater variations. hey are of opinion that there is some regularity in these variations. They could not observe a maximum value of Ra/r equal to 2. They found a maximum value equal to 0°73. Smoluchowski, assuming from the data of Zsigmondy available at that time that Ra/r=2, points out that the maximum rate of coagulation is reached when each collision between two particles is successful in bringing about a coalescence. When the rate of coagulation is slower, all the collisions are not successful in bringing about a coalescence of the particles. If ‘“‘e” is the fraction of the collisions that are successful in Meets about coalescence, then “ T” in equations (1) and (2) takes the form a No.” > ire ACRE c ti) ‘ o where No, Ra, 7, 0; and » have the same meaning as in equations (1) and (2). Putting 3 : Ng - 7) 1 (6) 4 Ra.€.n 8’ we have No _ eee ; 7 area ee. t (1) Leip eee OeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeEeeeeeEeEOEOeeeeEeeEeEeEeEeEeEeEeEeEeEeEeEe_oeG_ eee 308 Profs. J.N. Mukherjee and B.C. Papaconstantinou on Since only ‘‘e’’ is variable, a comparison of the coagn- lation time “‘t” for the same change in the sol makes it possible to determine the variation in the percentage of successful collisions and its dependence on the conditions of experiment. When the maximum rate is reached, e=1 and hence a measure of the absolute value of € is possible. Problems awaiting solution.— A 3zlance through the experi- mental work would show that the assumption of the constancy of “TT” is not well justified. The simplicity of Smolu- chowski’s equations consist in that there is only one constant. The experimental limitations are great, and it is quite possible that the discrepancies are due to the defects of the ultra- microscopic method. The other possibility is that the simplifying assumptions of Smoluchowski—for example, the constancy of “T”’ independent of the stage of co- alescence—are not true within narrow limits. It is of great interest to know the limits within which these equations are valid. The important questions that await solution in this connexion are : (a) the limits within which the above equations are valid ; and (b) if the above equations are valid, the variation of ¢ with concentration of electrolyte ; (c) the dependence of ¢ on the electric charge ; (d) the variation of e with temperature. In the following an account of an attempt to examine these factors, with the exception of (c), is recorded. Indirect Methods.—Variations in physical properties that occur simultaneously with the process of coagulation can be utilized to measure the rate of coalescence. Smoluchowski pointed out that the viscosity measure- ments of Gann (Koll. Chem. Beithefte, vol. vill. p. 67 (1916) ) do not satisfy the main requirements of his equations— namely, a similarity in the form of the curves (showing the variation in viscosity with time) independent of the nature of the electrolyte. He concludes that viscosity changes do not form a measure of the coagulation process. Yet he considers that the method is suitable for a quantitative comparison of the effect of various concentrations on the values of e when the curves are similar. The variation in physical properties, however, is likely the Kinetics of the Process of Coagulation. 309 to show the validity of the fundamental equations of Sinoluchowski. The fact that the curves showing the change in viscosity with time are dissimilar shows that these assumptions are not justified, and Smoluchowski thinks that “T” is dependent on the magnitude of the aggregates. Since as yet it is not possible to express physical pro- perties—e. g., the viscosity or the absorption of heht—in terms of definite functions of the number and size of particles, a quantitative comparison of different sols is not possible by indirect methods. We have, therefore, to restrict ourselves to the same sol. Experiments with Gold Sols.—An examination of the changes in the colour of gold sols on the addition of an electrolyte showed (Mukherjee and Papaconstantinou, Trans. Chem. Soe. vol. exvii. p. 1563 (1920)) that the variation in the absorption of light of old sols affords an easy and accurate method suitable for this purpose. The gold sols prepared by the nucieus method of Zsigmondy ‘conform very nearly to the requirements of equations (1) to (3) in so far as the particles are fairly uniform in size. It would be very convenient to work with a sol with re- producible properties, as data obtained on different dates with different preparations could be rigorously compared. It was found that a sol on standing for some time under- goes somewhat irregular changes, which may in part be due to dust particles cetting in accidentally. In spite of all precautions, one cannot be sure that there is no such variation in a particular sample. This variation is not wholly due to the fungus that grows in these sols. For this reason it is necessary to vary one factor only at a time and compare its effects. The comparison was therefore restricted to the same sol so long as it showed no variation in its properties. The Constancy of “T” in Equations (1) to (8) during the Process of Coalescence. According to the simple assumptiens of Smoluchowski, the progress of coalescence should be uniformly the same for various electrolytes and for their different concentrations. The constancy of * T” implies that if we assume a series of consecutive stages of coalescence of a sol—under a definite set of conditions, namely a definite electrolyte concentration 310 Profs. J.N. Mukherjee and B. C. Papaconstantinou on and temperature—following each other by intervals of time equal to “dt,” they are each characterized by a definite number and manner of distribution of particles of each category (primary, secondary, etc.). Let us indicate the stage of coalescence corresponding to the time ‘“t” seconds (since the sol and the electrolyte were mixed) under the given conditions by the numbers IN, No NANG ONG eee where the subscripts refer to the number of primary particles by the union of which the aggregate is composed. Thus N; denotes the number of aggregates, each of which is composed of “” primary particles. ‘“k” is evidently an integer. Similarly let us denote the stage of coalescence corre- sponding to the time ¢’ (=t+At) by > iN Ny’, Ne NGS ar ary N,,’ oe © e These stages of coalescence are independent of external conditions so long as equations (1) to (3) are valid. ‘Lhe only change that external conditions can bring about is a variation-in the value of T—that is, if the external conditions are varied the sol will always pass through the same consecutive stages of coalescence and only the rapidity of succession of these stages will be determined by them. Any property which varies continuously with the progress of coalescence without having any maxima or minima can be utilized to characterize the stages of coalescence ; for each value of this property is characteristic of the time that has passed since the mixing of electrolyte and sol. According to the equations of Smoluchowski, the times taken to reach any particular stage depend only on the value of ‘° T,”’ which is constant under a definite set of conditions. Let us compare two different electrolytes, A and B, of concentrations C, and Cy, Let us suppose that after the time ‘‘t”’ the stage ot coalescence indicated by > Nj NEN Gee yes he N ates has been reached when the electrolyte is “A” of con- centration C,. This stage of coalescence has a definite value for the physical property we are considering, and is independent of the value of T. Let us assumé@ that “T, ” and * T,” are the corresponding values of “‘'l’”’ for the two cases. ‘To be definite, we shall consider the variation in the the Kinetics of the Process of Coagulation. oii total number of particles of all categories, which varies continually with the progress of coalescence. Let us assume that at the times “?¢,” and “¢t,” both electrolytes have reached a state at which the total number of particles is the same. From equation (1) we have, therefore, No No Kk = = PN a ee AM 2) Sn ty ty (¢ ) I Ss m 1 ole mn 4 1 Dy t, e . nv. A . . . . . . 3) I’ Is ( ) The general equation (3) may be written as vw (BP Tei - Fae Since m and & are constants, if 7 is constant, n; has a (10) 2d 4 a ‘ag which is deduced I, I; from the condition that nn has a fixed value, also implies that the values of 71, no, nz, ... 7, are the same in both cases. This means, in other words, that a definite value of Sn fixes unequivocally the stage of coalescence. ‘Therefore, from the deduction that the successive stages of coalescence are always the same and depend only on the time, any property of the sol that varies continuously can be utilized to re- present a fixed value of Sn or n, or a definite stage of coalescence. A definite vaiue of this property is thus characteristic of the stage of. coalescence. It also follows from the above considerations that all curves showing a variation of this property with time should be similar. A deviation from this similarity, in itself, would mean that equations (1) to (3) do not represent the facts, The absorption coefficients of gold sols for different wave- lengths change on addition of an electrelyte in a complex manner. The theories of the colour of these sols as advanced by Maxwell Garnett (Phil. Trans. vel. cci. A, p. 385, 1904; vol. cev. A, p. 237, 1906) and by Mie (Ann. der Phys. [4] vol. xxv. p. 377) would lead one to expect that-any change in the number and manner of fixed value—that is, the condition ad 312 Profs. J.N. Mukherjee and B.C. Papaconstantinou on distribution of the particles n,, m», etc. will produce a great change in the optical properties of the sol. This is in agreement with observations. Now, if the successive stages of coalescence were independent of the nature and concentration of the electrolyte, then the manner of variation of the complex absorption would be the same in each case. The absorption in the red region of the spectrum varies continuously, corresponding to each value of the absorption coefficient for a particular wave-length in this region ; the values in the other parts should be fixed. If the contrary holds good, then the conclusion is obvious that the successive stages of coalescence are not inde- pendent of the nature and concentration of the electrolyte as assumed by Smoluchowski. It has been found that for the stage indicated by the value of the coefficient of absorption for 683 pu=0°4985, the values of the coefficient for the other wave-lengths given in the following table in column II. are independent of the nature of the electrolyte. The concentrations of the electrolytes were such as to produce rapid congulation. In columns III. and IV. the coefficients of the “nucleus sol”? have been given for the original sol and for the stage of coagulation cha- racterized by the value of the coefficient for 683 ppe=0°4156 (Mukherjee and Papaconstantinou, loc. cit.). TaBue I. Absorption coefficients (4). Wave-length, a Se oe ieee ae in pp I II. III IV O83 onuease: 0:0453 04985 0:0376 0°4156 OOD ye ares 0-1055 0°3679 O-1131 0°409 DSOs eee 0-3518 0:3388 0-1595 03986 DOS Ae 0:2076 0°3294 0°2076 0°336 OAT. haa 0:2512 0:3238 0°2867 03732 O23 3.0... | '0:3780 0°3780 0:3780 0°3882 DOG IER. 0:4647 0°3581 0°3882 0°3780 AO ee ree 0°3581 0°3198 03581 0°3780 Comparison of the Values of “T” as a Test of Smoluchowski’s Theory. Since the absorption coefficient in the red region varies continuously with the coagulation and its magnitude is the Kinetics of the Process of Coagulation. 313 sufficiently great, a definite value of the absorption co- efficient for a fixed wave-length (683 uw) can be taken as representing a definite stage of the coalescence. In the following tables the absorption coefficients at different times are given for the wave-length 683 yy. The tables are taken from the paper by Mukherjee and Papaconstantinou, loc. cit. TABLE TE Electrolyte: Potassium Chloride. Time in minutes, Absorption coefficients after mixing for various concentrations. equal volumes of - ——_—- aan electrolytes and sol. N/24. N/26. N/28. —- 0:0453 0°0453 0:0453 (CG See yee 0°3732 0°2867 0 1683 ee Sar es 0°438 0°3650 0:2257 fA ieee ane 0°4497 0°4046 — 1? ee eee — . 0488 0:2777 ete cia. -—— 04497 03431 EH ee ptt _ — 0°3836 + ae RE -- -= 0:4263 11) Ace ec — — 0°438 Ai 4 Seat — — 0:4497 TaBueE III. Potassium Nitrate. Concentrations. Times. — “~ ~ N/24. N/26. N/30. Se Ph aaek 0:0453 0:0453 0:0455 OPA ne 0°3336 — = i LAA She es 2 0°4263 0:2866 0269 ee eee 0:4497 0°3271 031438 Pe ag Ye ee — 0°3629 0°3356 Ee ae — 0°4156 0°3732 (A eee — 0°438 0°394 Tee? ~- 0:4497 — SMa ae — — 0°4263 WO ce ano es -- -—— 0:438 16 -— — 14497 314 Profs. J.N. Mukherjee and B.C. Papaconstantinou on TABLE IV. rb Chloride. Concentrations. Times ————= aS : = 0°852N/900. 0°852N/1000. 0°852N/1100. Se) 5. eae 0:0453 0°0453 0:0453 Ma aaicene 8 2 O°2257 — 0-1603 Ay Se MN ae 0°2867 — 0:2007 ZO ae ee 0°3529 — 0:2687 fy Re 0°3836 0°3051 0°3051 To ees 0438 0:3431 0°3237 Si aa 0:4497 — 0°3336 Oy Gian 0:4497 0°3836 0°3529 a: Gy Pees ee — 0°4263 0°363 [Sia eae = 0:4497 0:363 Gita dee ce: — — 0°3732 The limits within which the rate of coalescence could be varied were restricted by the fact that when the rate is slow the particles begin to settle, leaving a clear layer at the top, and the measurements are not reliable. Also, with time, some of the particles stick to the sides of the vessel. Lastly, it is difficult to avoid dust particles for a long time. The values given in Tables II.-V. were plotted graphi- cally, and the time intervals given in Tables V.-VII. below were determined from these curves. Hach of these curves is characterized by a definite value of T (ore). Corresponding to the three concentrations of any one of these electrolytes, there are three intervals which must pass in order that the absorption coefficient may have the same value. These intervals are co-related by the following relation according to equations (1) to (3) or (6) :— ain Oy ace aes ea SD, gee ie). (a) or bots: t; = 1): 2 le ey ees (12) Dalene Eee Since T,, T,, and T; are constant, the ratio of the time- parameters corresponding to the same absorption coefficient should be independent of the absolute value of the absorp- tion coefficient. Corresponding to different values of the absorption coefficient we get different values of ¢), ¢:, and fs. All these values should show a constant ratio. In the the Kinetics of the Process of Coagulation. 315 following three tables this comparison is made for the three electrolytes mentioned in Tables II.-IV. Values of absorption cvetlicient. Absorption coefficients. Absorption coeflicients. TABLE V. Electrolyte : Potassium Chloride. Time in seconds, —-oO; N/24. N/26. — ————— di N/28. a 255 sete er eeseoe TABLE VI, Hlectrolyte : Potassium Nitrate. Times. — pe N/24 N/2s. ie a a ogee 105 30 j or a 45 165 60 210 90 300 Extreme variation from average N/24. ~ 135 165 225 265 Extreme variation from average Average ey TABLE VII. Electrolyte: Barium Chloride. Times. —_ N/26. 277 345 430 480 Average eeeeee Peewee i 2°47 9:44 1 wei 2°) O's 1 Oe 3 13:0 il 274.--> 12-0 eee 11-4 14% 19° Ratios. Dune ey eee Dee We rae O42 4 he SSG) es Sain Nal Reese cor | se sOr2 Ieee. oor haeicao! | (OS. Bre 40 83,77 Ratios. OS hes Le il 2 POS, l 7g) eae. 1 IOs Dt 1 ist oO It will be seen from Tables V. to VII. that the agreement is as good as can be expected. The variation in T is as great as 1] times, but the ratios are constant. The agreement 316 Profs. J. N,. Mukherjee and B. C. Papaconstantinou on shows that the ratios of the values of T are independent of the time or the stage of coalescence. The ultramicroscopic measurements so far made show even during one experiment a much greater variation in ‘I’, as will be evident from the following tables :— Tasue VIII. (a). (Observer: Zsigmondy.) Values of #’= - Series D. Series H. Series F. 0-083 0°105 0-040 0-028 0-058 0:0195 00302 0-049 0:0183 0:0309 0:0475 0°0153 -—— 0:0403 0:0187 = — 0:0126 Zsigmondy used high concentrations of electrolyte for securing a rapid rate. When the rate of coagulation is slow and the duration of experiment is greater than a few minutes, he found that impossible values of ' are obtained. He thinks that the presence of impurities in the water used in diluting the sol for ultramicroscopic observations is the cause of this irregularity. In his case the miximum time covered by the experiments is 80 secs. Similarly, Westgren and Reitstétter, working with coarse gold sols, find the following range of variation in the constant :— Tasrn VITO). (Observers: Westgren and Reitstotter.) Values of a Series I. Series II. Series LILI. Series IV. O74 2:56 2°75 3°41 247 2°81 2°60 2°80 2°07 2°33 217 2°60 2:10 2°31 2°40 2°48 2°09 2°31 2°12 2°14 1°62 —- — — 1-41 2°16 2°15 2°15 a wy) — 2:05 VONAGE apehen opens 2-2 2°38 2°36 2°19 Extreme variation... 75 7% 10 % li % 55 7% the Kineties of the Process of Coagulation. 317 Kruyt and Arkel *, working with selenium sol and very slow rate of coagulation, find extr emely wide variations in T’ in the same exper ut PAsnE IX: (Observers: Kruyt and Arkel.) Values of T (in hours). if Ti al IV. 28 260 131 1:3 51 390 55 3-4 44 270 52 2:2 (43) 320 54 43 (157) 600 68 10°5 200 sey (0 55 40 a 510 a * = 440 48 37 ae = = 52 : af a 38 The above few instances will suffice to show the range of variations in ** T” during the course of one experiment that has been observed in the ultramicroscopic measurements. Considering that in Tables V. to VII. the ratios between the different values of T are taken, the range of variation is extremely small. The actual deviations in the value of T in any one experiment must be much less than the extreme variations given. ‘This comparison leaves no room for doubt that “T” is a constant in the case of gold sols and within the limits of the rate of coagulation that have been studied. In fact, these data constitute the best evidence so far recorded in favour of the theory of Smoluchowski. The Dependence of € on the Concentration. Tables V. to VII. show clearly how rapidly e, the percentage of successful collisions, increases with con- centration. A change of concentration in the ratio 24 to 28 increases the rate in the ratio 1:11 or 1:6 as the case may be. It wouid be extremely interesting to work with a sol which is less susceptible to impurities tan these gold sols. * Ree. Trav. Chim. Pays-Bas, vol. xxxix. [4] p. 656 (1920); [4] vol. xl. p. 169: (1921). 318 Profs. J.N. Mukherjee and B. ©. Papaconstantinou on Variation of T or € with Temperature. ) Similarly, by determining the times required to produce a definite change in the colour of the sol for the same electrolyte concentration but different temperatures, we can determine the variation in e with temperature. From equation (3), pail No . ae we Nt we get No pe ~=1 Oct. i Se Sune (13) Since a definite change of colour is being used, a is constant, or . 8 14+ 8 .¢,.¢ = %), 2 constant. “2..7) aaen, Substituting the value of 6 in (14), we get 4 Ra ° d - No ~ I+3 No.7 INS 0 : ° . > (Gi, Since Ra, No, and zp» are constants, we have fb . G 1) The viscosity of colloidal gold solutions has been found to be practically equal to that of water, and the variation with temperature can be assumed to be equal to that of water. For different temperatures we have = k'5 a Constant, (A... 6); oo eal iy JOnees (tty Ce ves hg a eee 1) UP Sie Since ¢, is experimentally determined and @ and 7 are known, variations in € can be compared. The experimental data are given below. They are taken from the same paper (pp. 1570-71). TABLE X. | Temperatures. — aS — Electrolyte. Standards *, Le. 30°. 50°. N/30 Potassium chloride... V Sol.C. Simin. 10 min. 8 min. 80 see. N/30 Potassium sulphate. _,, 3 WL 6 dO'sec.’.. 10 see: 10 sec. N/30 Potassium nitrate .. ,, NG Dae: any Wh 55 Leas * These refer to the protected gold sols used as standards for comparison of colour. See Zoc. cit. the Kinetics of the Process of Coagulation. 319 TABLE XI. Electrolyte: Barium Chloride. Sol. E. Temperatures. Con- — on eS centrations. Standards, 15°. 30°. 40°. 50°, 0°852 N/1000 ve Tmin. 6min. 4min.50ser. 4 mir.:20 sec. 0°852.N/1000_ ~—s: BB, 34 ,, BS\. a> — Tati oO 0°852 N/1200 Mie Lae US ee 12 min. 30 sec. Gee? Lo. |; 0°852N/1200 3B, 124 ,, Oe) 333 62 min. or TasLe XII. Electrolyte : Strontium Nitrate. Sol. F. Temperatures. Con- = i eis ve centrations, Standards. Lae. 30°. 50°, N/1000 V; 1 min, 10 see, 20 sec. 8 sec. N/1000 By S.xie nib be 1 min. 40 see. 45 ” At 15°, 30°, 40°, and 50°, 7/0 has the values 3°96 x 10~°, wert), 2-1 10-5, and “1°7 x 10~ ‘respectively. The values for the viscosity are taken from the tables in Kaye and Laby’s book on Physical and Chemical Constants, peau; 1919. Krom equation (17) we have Ey50 + Exq0 * Ego * €509 = (/t@) 15° s (7/t@) 30° (7/tO) 40° 5 (/t®) 50°. TasLEe XIII. em peratures. Electrolyte. 5°. 39°. 50°. NSO KG) Yas.'ss083 N/t@ x 107 1-32 0°50 0°33 Bi a0 WSO) 9 5.22. aN 13:2 35 ly W/o) KNO, 2.32: % 9:43 184 14:0 TasLe XIV. Electrolyte: Barium Chloride. Temperatures. » Con- pa eee centrations. Standards. LOTS eee Oe.” BAS: 0°852 N/1000-...... VE n/t0 x 108 9°45 972 7:24 6:54 ea Suet. B, ‘3 1°94 2:4 -- 2] Ratio between { ae | eer pepe Ieee TT i 70 ID os caiccerey al ihe se dak 100 52). : 108 0°852 N/1200 .. v. n/t@ x 10° 2°87 4:24 2:8 4:5 53 Par tot By; rt 53°22 745 64 — Ratio between { 2 Bera eee 1cGne> ie : 98 : 158 TL a, Oe BEEGy ‘bigs ecesccus Wes 140; 106 : — 320 The Kinetics of the Process of Coagulation. TABLE XV. Hlectrolyte : Strontium Nitrate. Temperatures. Concentrations. Standards. 150. aan 50°. INT NOOO, cocen ces V, /t@x 106 "965 1°65 271 SS: ae re B, /t0x 107 “80 3°93 O77 atio waiween: (vie eeeenee 1007 2 290 ou iy COMM eee | Boece ee 100 : 410 : 480 Since 7=¢6 is a constant for a definite electrolyte con- centration and temperature according to Smoluchowski’s equation, the ratios should be independent of the standard used. This is true within the limits of experimental error with -852 N/1200 Barium Chloride. In the other two cases the variations are not great considering that we are com- paring the ratios. A slight variation in each value will be magnified in the ratio. Taking into account the probable experimental error, it can be said that eis roughly constant in each experiment. On the other hand, the variation in e with temperature is considerable. We have already seen that the irreguiarity in the variation of e means that the precipitating power of © the ions changes with the temperature (Mukherjee, Trans. Chem. Soc. vol. cxvii. p. 358, 1920). Further experiments with arsenious sulphide are in progress on similar lines. Summary. (1) It has been shown that the equations of Smoluchowski on the rate of coalescence of the particles of gold sols agree with the results obtained by the writers. (2) It has been suggested that the disagreement of the ultramicroscopic measurements with this theory may in part be due to the difficulties inherent in them. Our best thanks are due to Professor F. G. Donnan for his kind interest and encouragement, and also to our friend, Professor J. ©. Ghosh. Physical Chemistry Department, University College, London, XXIX. The Adsorption of fons. By JNANENDRA Natu MUKHERJEE, D.&c., Professor of Physical Chemistry in the University of ¢ jale GED >. + a paper in the Transactions of the Faraday Sceciety (Far. Soc. Dise. Oct. 1921) an attempt has been made to define the nature of the adsorption of ions to which the origin and the neutralization of the charge of a colloidal particle are due. The origin of the charge was assumed to be due to the adsorption of tons by the atoms in the surface as a result of their chemical affinity. It was pointed out that the adsorption of one kind of ions will impart a charge to the surface, in virtue of which ions of opposite sign will be drawn near the surface. In the liquid there remains an equivalent amount of ions.of opposite sign. The electrical energy will be a minimum when these ions are held near the surface so that the distance between the oppositely charged ions has the minimum value possible under the conditions, and they will be held opposite to the ions chemically adsorbed. An “ion” so held will not be “free” to move if its kinetic energy is less than “ W ” the energy required to separate the ion from the oppositely charged surface. The number of such “‘ bound ” ions deter- mines the diminution ia the charge of the surface. When the concentration of ions of opposite charge in the liquid is small the number of ions “ held” to the surface by electrical attraction will be small. If the chemically adsorbed ions have a valency equal to € N,,” and if ‘ N,” is the valency of the oppositely charged ions in the liquid in contact with the surface, then where H = the electronic charge, « = the distance between the centres of the ions at the position of minimum distance, and ‘* D)”’ is the dielectric constant of water. Depending on the concentration of the oppositely charged ions in the liquid near the surface, at any instant a certain number of the “chemically adsorbed” ions are ‘* covered ” by ions of opposite charge. In the liquid near the surface there are always a number of free ions equivalent in amount to the “ uncovered’’ chemically adsorbed ions on the surface. The total amount of ions of opposite sign both “ bound”’ and * Communicated by Prof. F. G. Donnan, F.R.S. Phil. Mag, Ser. 6. Vol. 44. No. 260. Aug. 1922. od 322 Prot. J. N. Mukherjee on ‘“‘free”’ is equivalent to the amount of ions ‘ chemically. adsorbed.” These “‘ free”? ions form the second sheet of the double laver. It is evident that as a result of their thermal motion the mean distance between the two layers will be greater than “2.” The charge of the surface was treated as due to discrete charged particles widely separated from each other compared with molecular dimensions. [It was shown in the previous paper that this view gives a rational explanation of the fact that a reversal of the charge of a surface can be brought out only by polyvalent ions of opposite charge. The equilibrium conditions were iieenesed and the equa- tions deduced were shown to be in agreement with the valency rule, the influence of the mobility of the oppositely charged ion, and with the influence of concentration on the charge of the surface. Oniv the theoretically simplest case has been discussed in the earlier paper. In the present paper the more important facts connected with the adsorption of ions are discussed from this point of view, and it will be seen that this view gives a simple explanation of most of the general conclusions already arrived at on experimental grounds. Theories of Adsorption. Before proceeding to discuss the adsorption of ions it will be convenient to deal briefly with the different views advanced to account for adsorption in general. The with- drawal -of a solute from a solution by a solid may be the result of the formation, of definite chemical compounds, of solid solutions, of mixed crystals and surface-condensation. In many cases all these changes are simultaneously present. In this paper the word “adsorption ” denotes condensation or combination, at the surface only, without the interpenetra- tion of the adsorbed substance throughout the mass of the adsorbent (Mecklenburg’s criterion, Ze Phys. Chem. \xxxuii. p. 609 (Lona ae: also the sense in which the term is used in deriving Gibbs’s equation). Faraday (Phil. Trans. exiv. p. 55 (1834)) in his well-known explanation of the catalytic combination of hydrogen and oxygen on platinum surfaces, remarks “‘that they are de- pendent upon the natural condition of gaseous elasticity combined with the exertion of that attractive force, possessed by many bodies, especially those which are solid, in an eminent degree, and probably belonging to all, by which they are drawn ks aaaooian fan quore on. lee close, sulinemi at ae same time undergoing chemical combination though often the Adsorption of Lons. Sea assuming the condition of adhesion, and which occasionally leads under very favourable Genre as in the present instance, to the combination of bodies simultaneously sub- jected to this attraction.” It is remarked further “that the sphere of action of particles extends beyond those other particles with which they are immediately and evidently in union, and in many cases produces effects rising into con- siderable importance.” These remarks of Faraday mean, in modern terminology, that there is a sort of combination at the surface and that the transitional layer is more than one molecule thick. The subsequent views are in a way develop- ments of this conception. | Gibbs treated adsorption from the standpoint of thermo- dynamics. A number of important investigations has been earried on by Milner (Phil. Mag. [6] xiii. p. 96 (1907)), Lewis (Phil. Mag. [6] xv. p. 506 (1908)), sbed. xvii, p. 466 (1909)), and Donnan and Barker (Proc. Roy. Soe. Ixxxy. A. p. 552 (1911)). The present position is that the amount adsorbed is often considerably greater than what could be expected from Gibbs’s equation. J.J. Thomson (‘ Applications of Dynamics to Physics and Chemistry’) showed that it follows from Laplace’s theory of capillarity that in the surface layer between two liquids, chemical actions may take place which are absent in the bulk of the liquids. Lagergren (Bihang K. Svenska Vet. Hand. xxiv. p. 11, No. 415 (1898)) considers that adsor ption in the surface of solids in contact with aqueous solutions is due to the com- pressed state of the water in the surface layer. On the experimental side the work of Freundlich and his collaborators—|[ Kapillar-Chemie, 1909; Z. Phys. Chem. lix. p. 284 (1907); Ixvii. p.538 (1909); Ixxiii. peogs) (1910) ; ixxxill. p. 97 (1913); Ixxxv. p. 398 (1913); men py bol (1915); Koll.-Chem. Beihefte, vi. p. 297 Rees) : see also Schmidt, Z. Phys. Chem. \xxiv. p. 689 (1910) ; Ixxvii. p. 641 CEPEL). 3 fxxviil. p. 667 (19%); lxxxiil. p. 674 ( (1913); xci. p. 103 (1916). In the last-mentioned paper Schmidt and Hinteler conclude that Freundlich’s equation represents their experi- mental data better than that of Schmidt |—and of others, have shown that adsorption-equilibria can be generally expressed in terms of the well-known equation of Freundlich: fy ee 1/ 2 Bim Ce 2! ew Ce) Freundlich expressed the opinion that adsorption is mainly due to a decrease in surface tension as suggested by Gibbs. ¥ 2 324 Prot. J. N. Mukherjee on In the case of adsorption of gases by solids, Arrhenius (Medd. f. k. Vef. Nobelinstitut, ii. N. 7 (1911); Theories of Solution, 1912, pp. 55-71) has drawn attention to the parallelism between the van der Waals’s coefficient “a” for the different gases and the amounts of these gases dee by charcoal, and he believes that this is deans evidence of the compressed state of the surface layer. At the same time . he lays stress on the chemical aspect—namely, that in addition to the attractions between the molecules of the gas in the surface layer, one has to consider the chemical attraction of the surface atoms and the molecules of the gas. Recently, Williams (Proce. Roy. Soe. xevi. A. p. 287 (1919) ; xevil A. p. 223, (1920); also Trans. Mar. Soc. x.) glee (1914), in which complete references to the literature on negative adsorption are given) has treated adsorption from the points of view of Lagergren and of Arrhenius in a number of interesting communications. It may be mentioned here that the disagreement of ob- servations with calculations from Gibbs’s equation is at least. in part due to the fact that only one source of change in the free energy of the surface layer is taken into account. In the simplest case ot the interface, liquid-saturated vapour (one component system), it is open to objection whether “vy” denotes the total change in free energy of an isothermal and reversible-formation of unit surface. Bakker (7. Phys. Chem. Ixvui. p. 684 (1910)) has pointed out that if the density of the surface layer is different from that of the liquid in bulk a second term is necessary to represent the change in free energy. It is possible that in this particular case this second term is negligible in comparison with “‘y,” the tension os unit length at low temberatures, but at hai temperatures « ” has a low value and the saturation pressure is very pee so that the second term may be even more important. Williams (Proc. Roy. Soc. (Edinburgh), xxxviii. p. 23 (1917-18) ) has drawn attention to the effect of the variation of the ee of an adsorbent when adsorbing—a factor which is very often neglected. Lewis (Z. Phys. Chem. |xxui. p. 129 (1910) ; also Par- tington, ‘Text-book of Thermodynamics,’ p. 473 (1943) has discussed the influence of a variation in the electric density on the surface on the form of Gibbs’s equation. These may be called the physical theories of adsorption. The difficulty in accepting them as general theories of ad- sorption is that they attempt to explain adsorption in terms of a single physical pce e.g. diminution in surface energy the Adsorption of Lons. 329 or a layer under great internal pressure. The necessity for recognizing the existence of a sort of chemical interaction (as Arrhenius has suggested) becomes evident when one con- siders the specific nature of adsorption processes. This point has been justly emphasized by Bancroft in recent years. Besides his papers in the ‘Journal of Physical Chemistry,’ compare ‘ Applied Colloid Chemistry,’ 1921, p. 111). The chemical point of view has been pe clearly by Lang- muir (J. Amer. Chem. Soc. xxxvili. p. 2221 (1916) ; RIK p. 1848 (1917)). He believes that adsorption is due to the chemical affinities of the surface atoms. Considering the thermodynamic equilibrium between molecules of a gas at the surface and those in the surrounding gas he has deduced the following equations correlating the variation of the ad- sorbed amount with its pressure, where ‘‘ @,”’ is the fraction of the solid surface covered and is a measure of the amount adsorbed, v, is the rate at which the vas would evaporate if unit area of the surface were completely covered, “‘w” is the number of gas molecules striking unit area of the surface per second and is given by o=—43'15 x10 aT A and ‘“‘»’’ denotes the pressure of the gas, ‘‘T” its absolute temperature, and ‘“‘M” its molecular weight. « denotes the fraction of the total number of collisions of the molecules of the gas that leads to a condensa- tion on the surface; it is usually close to unity and evidently can never exceed unity. Some Interesting applications of his theory to catalysis of gaseous reactions by solid surfaces are given, This theory explains many phenomena which are otherwise difficult to understand. Michaelis and: Rona (Bio-Chem. Zeitsch. xevil. pp. 56, 85 (1919)) conclude from the investigations of Michaelis and his co-workers that the assumption of special forces at the sur- face fails to account for the facts and that adsorption is the result of chemical affinity. I. TheAdsorption of a Constituent Ion by a Precipitate. The adsorption of ions is different from the adsorption of neutral molecules or groups in that it introduces a new factor—an electrically charged surface. The variation in the electric charge enables us to follow the net effect of the adsorption of the two ions, as the electric charge depends 326 Prof. J. N. Mukherjee on only on the total number of ions (of both signs) fixed per unit area of the surface. Kataphoretic and electro-endosmotic experiments give us a quantitative idea of the relative ad- sorption of both ions. _ The electric charge helps to peptize the adsorbent, and a qualitative idea of the adsorption of ions can be formed from peptization by electrolytes. An insoluble precipitate formed by the union of two oppositely charged ions has a marked tendency to adsorb its component ions. In many cases the connexion between the adsorbed ion and the electrical charge has been established. These instances have been given in the earlier paper. The nature of the chemical] ’ forces responsible for this adsorption has also been defined. Instances of adsorption of ions as judged from peptization by electrolytes are given below. Bancroft (Rep. Brit. Assoc. p. 2 (1918)) remarks :—‘ It seems to be a general rule that insoluble electrolytes adsorb their own ions markedly, consequently a soluble salt having ene lon in common with a sparingly soluble electrolyte will tend to peptize the latter. Freshly precipitated silver halides are peptized by dilute silver nitrate or the corresponding potassium halide, the silver and the halide ions being ad- sorbed strongly. Many oxides are peptized by their chlorides and nitrates, forming so-called basic salts. Sulphides are peptized by hydrogen sulphide. .... The peptization of hydrous oxides by caustic alkali can be considered as a case of adsorption of a common ion or as the preferential adsorption of hydroxylion. Hydrous chromic oxide gives an apparently clear green solution when treated with an excess ot caustic potash ; but the green oxide can be filtered out completely by means of a collodion filter, a colourless solution passing through.” “Hanztsch considers that hydrous beryllium oxide is peptized by caustic alkali, copper oxide is peptized by con- centrated alkali, and so is cubalt oxide. In ammoniacal copper solutions part of the copper oxide is apparently colloidal and part is dissolved. Freshly precipitated zine oxide is peptized by alkali, but the solution is very unstable ” (cp. also ~ negative hydroxide sols— Freundlich and Leonhardt, Koll. Chem. Bethefte, vii. p. 172 (1915)). At least in some of these cases the formation of new com- plex anions is possible, and it is not definitely known to what ion the peptization is due. Regarding the peptization of stannic acid gel by small quantities of alkali, Zsigmondy (Kolloidchemie, p. es et seq. (1920); also Varga, Koll. Chem. Bethefie, xi. p. 26 (1919)) remarks: “Dieses kann the Adsorption of Lons. 327 sowohl auf Adsorption des gebildeten Katiumstannats wie auch darauf ziiruckfiihren sein, das Kaliumhydrat mit den Oberflichenmolekeiilen der Zinnsiureprimiirteilchen in Reaktion tritt, wobei diese von der Oberfliiche der Primiir- teilchen festgehalten werden.” The view suggested by the writer to account for the ad- sorption of a common ion, leads one to expect that ions which can displace one of the constituent ions in the crystal lattice should also be adsorbed. Mare (Z. Phys. Chem. Ixxx1. p- 641 (1913)) has observed that crystalline adsorbents adsorb erystalloids to any marked degree only when they can form mixed erystals with them and are isomorphous with them. Paneth and Horrowitz (Physik. Zeitsch. xv. p. 924 (1914)) have noticed that of the radio elements those only will be adsorbed that can form insoluble salts with the common ion of the adsorbent and can also form mixed crystals with the adsorbent. This kind of adsorption is somewhat different from the type we have considered for, as Paneth has pointed out in his case, an actual interpenetration of the two non- common ions is occurring in the crystal lattice. Thus radium is taken up by barium sulphate giving out to the solution barium ions in exchange. Such an interthange will not impart a charge to the surface. Attention may also be drawn to the explanation advanced by Bradford (Biochem. J.x. p. 169 (1916); xi. p. 14 (1917) ) to account for zonal precipitations, first studied by Liese- gang. Bradford thinks that the adsorption of a constituent ion is responsible for their formation. From the numerous instances given above, this conception seems to be quite plausible. It is probable that other factors have also an influence on the process (Hatschek, Brit. Assoc. Rep. p. 24 (1918)). Il. The Variation of the Density of the Electric Charge with the Concentration of an Hlectrolyte. In the previous paper the particular case when the charge of the surface is due to strong chemical adsorption of ions of one kind and when the added electrolytes have not any ions, subject to the chémical affinity of the surface atoms, has been fully treated. In this case it was assumed that the number of ions adsorbed at the surface by chemical affinity remains constant. The experimental data of KElissafoff on glass and quartz agree well with equations deduced from these assumptions, on the basis of the theory of . electricat adsorption. 328 Prof. J. N. Mukherjee on The general case, however, is that : (a) At low concentrations the density of the charge on the surface at first increases to a maximum and at higher concentrations falls gradually towards a null value when the oppositely charged ions are monovalent. (b) On the other hand, when the oppositely charged ions are multivalent or complex organic lons the charge passes through a null value, becomes reversed in sign, and again reaches a second maximum, after which it falls slowly (Hillis, 7. Phys. Chem. Ixxvit p. 621 (A911) > lxxxo pea (GEO ix x aie P 145 (1914); Powis, Z. Phys. Chem. Ixxxix., pp. 9, 179 (1904) 3" Riety, ‘Compi.” wena netme pp- 1411, 1215°(1912)5 ela p. 1368" (1913) > Youre ame Neal, -/. Pigs. Chem: xxiep. a UGT is akGrny te Verst. Kom Akad. v. Wetensch. Amsterdam, 27th Juin, 1914, also Koll.- Aevesch. XXile p. wed (VILS)). The usual explanation is as follows :— The adsorption-isotherms for the two ions can be written as | Ds ee and aye, we where the subscripts A and K refer to the anion and the cation respectively. To explain the increase in the charge at low concentrations it has to be assumed that as OAK and Ba < foure aT sto de ° (4) Thus in a paper read at the Discussion on Colloids arranged by the Faraday and the Physical Societies of London, Svedberg remarks: “* Now as a rule, it happens that for the two ions of a salt both a and 8 have different values, e.g. x (cation) <« (anion) B (cation) > B(anion).” It is clear that the equation of the adsorption-isotherm can be reconciled with the first increase in the charge. But two objections can be raised against this empirical point of view. In the first place, no reason is given why the constants « and B shall have generally the relative values assumed above for - the cation and the anion. Secondly, these assumptions can- not explain the second maximum charge and the subsequent decrease. observed with multivalent ions of opposite charge. It will now be necessary to assume that CL ier and Ba>Bx, . ahi ae . (5) in direct contradiction to the assumptions already made the Adsorption of Lons. 329 (ep. (4)). Besides, one cannot get any idea as to why the anion is generally more strongly adsorbed at low con- centrations. The facts ean, however, be explained as follows :— The negative charge of surfaces in contact with water is to be sought | for in the chemical natures of the anions and the cations. The simpler electrolytes (excluding dyes and complex organic ions) have cations whose chemical behaviour can be referred simply to the tendency of the component atom (eg. of the alkali and alkaline earth metals) to pass into the ionic state. These ions do not form any complex ions. They form only one type of compounds that are stable in aqueous solutions, namely, electrolytes with the atem existing as a positively charged ion through the loss of one o or more electrons. On the other hand, the anions in general co) form types of compounds other than electrolytes, and also _form complex ions. It is, therefore, possible to imagine that anions are subject to the chemical affinity of the surface atoms and that the chemical action on the cations is relatively small. Complex cations like those of the basie dyes should, for the same reason, be easily adsorbable. ‘This isa wok known fact. ; If now, the assumption is made that the chemical affinity acting on the anion of the electrolyte added is stronger than the electrostatic attraction of the surface on the cation, the observed variation of the charge with the concentration of the electrolyte is easily accounted for. This case corresponds to a strongly marked maximum of a negative charge at a low concentration of the electrolyte. The initial charge of a surface in contact with pure water can be due either : (a) to the strong adsorption of an ion of a minute quantity of suitable ‘electrolyte associated with the solid, (b) or to the adsorption of hydroxy] ions from water. On the addition of an electrolyte the density of the electric charge will increase at low concentrations because of the chemical adsorption of the anion. The electrical adsorption of the cation is smaller as the chemical adsorption has been assumed to be stronger. Besides, the electric charge of the surface is also not at its maximum. As the surface becomes more and more covered by the anions the rate of adsorption da/de—where ‘‘ dx” is the increase in the amount adsorbed per unit surface due to an increase in the concentration ‘‘de”’—rapidly decreases. Also, the electric charge repels the anions, and those only can strike on it that have sufficient 330 Prof. J. N. Mukherjee on kinetic energy to overcome the potential of the double layer. The number of collisions is thus not proportional to the concentration but rises more slowly. Near about the point where the surface becomes saturated the value of da/de will be almost zero (cp. the shape of the adsorption-isotherms. of Freundlich, Arrhenius, and Langmuir). On the other hand, the electrical adsorption increases continually with the concentration and the increase of the charge. It is. apparent that soon a balance will be reached between the chemical adsorption of the anion and the electrical adsorption of the cation. The minimum charge will correspond to the stage when da/dc for the cation is just equal to da/de for the anion. Beyond this concentration the charge will decrease rapidly, and when the surface has been saturated with the anion the subsequent variation in the charge is simply due to electrical adsorption. ‘The reversal of the charge by electrical adsorp- tion has been discussed in the earlier paper. It is necessary to add that as the electrically adsorbed polyvalent cations impart a positive charge to the surface, the atsorption of the cation decreases and the electrical adsorption of the anion — becomes possible. As long as there is a positively charged surface the adsorption of the anion will increase more rapidly with the concentration than that of the cation. A second maximum will thus be reached and a decrease in the charge will follow. The electrical adsorption of the anion is small because of the smallness of the positive charge and an initially existing negatively charged surface. A further reversal of the charge is not possible, and, in fact, has never been observed. TI. The Action of Acids and Alkalies. The works of Perrin and of others (J. Chim. Phys. i. p- 601 (1904); ii. p. 50 (1905); Haber and Klemensie- wiez, Z. Phys. Chem. \xvii. p. 385 (1909); Cameron and Oecttinger, Phil. Mag. [vi.] xviii. p. 586 (1909)) have shown that hydrogen and “hydroxyl ions behave exceptionally in that they impart to the surface a charge of the same sign as they carry. This behaviour is in contrast to that of the other univalent ions. Perrin attributes their singular activity to the smallness of their radii. In order to explain the presence of these ions, in excess, in the surface layer, it is necessary to assume some sort of a restraining force acting on them at the surface. Haber and Klemensiewicz consider that there is an ad- sorbed Jayer of water in the surface by virtue of which the the Adsorption of Ions. 331 solid acts as a sort of combined hydrogen and oxygen electrode. They treat the subject from the points of view of thermodynamics and Nernst’s theory of electrolytic solution tension. It has been pointed out by Freundlich (and Rlissafoff, 7. Phys. Chem. Ixxix. p. 407 (1912)) that hydrogen and hydroxyl ions are not the only ions which impart a charge to the surface. In many cases, acids have been observed not to reverse the charge at all. Many sub- stances have a negative charge in contact with pure water. These facts show that selective adsorption of hydroxyl ions has also to be considered. This thermodynamic treatment from the point of view of Nernst’s theory does not attempt to explain electro-endosmosis. For this purpose it is necessary to conceive of an electrical double layer, of which the layer imparting « charge to the surface is fixed relative to the mobile second layer. Freundlich, and Freundlich and Rona (Koll. Zeit. xxviil. 5, p- 240 (1921); Kol. Preuss. Akad. Wiss. Berlin, 1920, p- 397, C. 1920, ii. p. 26) have shown that the potential measurements by Haber’s method are not in agreement with those measured by electro-endosmotic experiments. They therefore suggest that there are two distinct drops in potential as one passes from the solid to the liquid (glass to water). The first drop is wholly in the solid and is probably of the nature associated with the Nernst theory of electrolytic solution-tensions. The second drop is in the liquid and composes the Helm- holtzian double layer which it is necessary to assume to explain electro-osmosis and cataphoresis. At the same time the characteristic effects of hydrogen and hydroxyl ions on neutral substances like barium sulphate, silver chloride, naphthalene, etc., point strongly to the correctness of Haber’s fundamental assumption that the explanation is to he sought in the equilibrium between the hydrogen and hydroxyl ions in the adsorbed layer of water and those in the bulk of the liquid. Williams (Proc. Roy. Soe. xeviii. A. pp. 223 (1920)) has recently suggested that the layer of water adsorbed on a charcoal surface is under great internal pressure (about 10,000 atmospheres). Applying Planck’s equation he shows that the effect of this pressure will be to increase the con- centration of hydrogen and hydroxyi ions in this layer. This increased concentration will set up a diffusion potential. He draws attention to the difficulties in accepting this view of the origin of the potential difference at the surface. In the cera considered by Haber and Perrin, the solid has a32 Prof. J. N. Mukherjee on little or na potential difference in contact with pure water, and the considerations developed by Williams are not applicable. Case 1.—The surface 1s inert. We shall assume that the atoms in the surface do not — exert any chemical affinity on hydrogen and hydroxyl ions as such, or on the dissolved acid (or alkali) with which it may be in contact. The adsorbed water molecules behave as a solid layer, being held by strong chemical forces (Haber, loc. cit.; Hardy, Proc. Roy. Soc. lxxxiv. B. »..217 (aia It is clear that the surface will be neutral in contact with pure water. The molecules of water in the adsorbed layer are in thermodynamic equilibrium with those in the bulk of | the liquid. It is reasonable to imagine that a transfer of an electron is taking place between the hydrogen atom and the hydroxyl group in the water molecules in the surface layer, as it does in the molecules in the liquid. That is, the water jolene are dissociating into ions at a. definite rate. Let “ne? be the number of water molecules (in the adsorbed ye passing into the ionized phase per unit area per second. For equilibrium, as many hydrogen and hydroxyl ions are uniting to form neutral water molecules. Since the adsorbed water molecules behave asa solid layer, recombina- tions would take place mostly between adjacent hydrogen and hydroxyl ions. The recombination will be extremely rapid. It can be assumed that at any instant the number of hydrogen or hydroxyl ions actually remaining free in the surface will be a negligible fraction of the total number of water molecules. The neutralization of the ions being formed in the surface layer can also be brought abont by impinging hydrogen or hydroxyl ions present in the liquid. In contact with pure water the probahility of such collisions is small, for the concentration of hydrogen and hydroxyl! ions is extremely — small. Thus neutralization of the ions being formed in the surface layer is possible in two ways : (1) H,° + OH,!—» HOH—the subscript “s” refers to ions in the surface layer ; (2))a(@ elo oe Ott el Ot, (>) HP + OH,—+ HOH—the subscript “f” refers . to the freely moving ions in the liquid. In contact with pure water, neutralizations according to scheme 2 are small in number. Also 2 (a) and 2(b) are equally probable. Consequently the numbers of H,° and OH," remaining in the surface at any instant will be equal, and the surface will be neutral. the Adsorption of Ions. 333 When an acid is added to the water the neutralizations according to scheme 2 (a) will be Sener negligible, but those according to scheme 2(6) will not be so. The total number of neutral molecules of water formed in the surface is still equal to “a,” but a number of them is now being formed according to 2 (6). Corresponding to the number of neutralizations according to 2(b), a number of hydrogen ions will remain in the surface layer in excess of the number of hydroxyl ions. The rate at which 2(b) proceeds thus determines the free charge on the surface. An equivalent number of anions remain unneutralized in the liquid and form the second mobile sheet of the double layer. The free charge on the surface will evidently increase with rise in the concentration of hydrogen ions in the solution. There are, however, two factors opposing this increase in the charge of the surface. A. The proportion of hydrogen ions striking on the surface diminishes as the positive charge of the surface increases. Only those ions which have sufficient kinetic energy to over- come the electrical repulsion can reach it. If e€ be the potential of the double layer in C.G.S. units, then the number of collisions of the ions per unit surface per second is pro- portional to Uno : Cro - e-€-B/KT, Po Vile aie tiene (6) where Cyo denotes the concentration of free hydrogen ions in the liquid, ‘‘ Hi” is the electronic charge in C. GS. units, T is the absolute ae ture, Ugo is the mobility of the hydrogen i ions in water, ana K =R/No, where “ R” is the gas constant and No the Avogadro number. - B. The other factor that tends to diminish the charge of the surface is the electrical adsorption of the anion of the acid added to the solution. That this plays an important part will be evident from the following examples taken from the observations of Perrin :— Rate of Electro-- Substance. Electrolyte. endosmotic outflow. PASO, coca t<~ M/1000 HCl. + 110 2 RAGAN Ae M/1000 citrie acid + 5 Pee Perey M/1000 HNO; (or HCl) + 100 APEC enere M/1000 H.SO, + 15 CO re us vii 0 M/1000 HNO; + 85 jee, s. M/1000 H.SO, + 2l jin Rees M/500 HCl + 90 Pete M/1000 H.C.0, + 30 as tea Feebly acid with HCl + 79 Pibege he: Solution of KH. (POx:) ) with approximately the | ete 4 same number of free hydrogen ions as above —— Oe SS eee ee ; ; O34 Prof. J. N. Mukherjee on Both these factors tend to diminish the rate of increase of the charge with rise in the concentration of hydrogen ions. For acids with simple univalent anions, the electrical ad-. sorption at low concentrations can be left out of account in view of the excessive mobilty of the hydrogen ions. A quantitative relationship can now be obtained between the charge on the surface and the concentration of the acid. Let w! be the rate of neutralization according to 2 (b) above. We have then wi=K ia. Cy. 76: EAT, oy ea (7) where & is a constant. ‘The density of the charge on the surface is proportional to «'—which is a measure of the number of hydrogen ions remaining in excess in the surface. If the thickness of the double layer remains constant then the potential of the double layer is proportional to the density of the charge: HOVENG Ik NO) aban When all the hydroxyl] ions in the surface layer are being neutralized according to 2 (6) the surface will have a maximum charge determined by ‘‘ w.”’ Putting «/z=@, since “x” is a constant, we have ea 3. and @ represents the ratio of the hydrogen ions present ‘in excess at any instant in the surface layer to the maximum number possible when the neutralization takes place only according to 2(b). The potential of the double layer can be written as exh. a'=h,.0=ks.Cqo.e78-%?. Uno, . . (9) or) OS ky: Ono.€ 70 YU 4 kr where hy, ko, ks, ko and 8 denote constants. Siailarly, for alkali solutions we have O=hy.Com.e 8-9/2, Ucn. A ee hc (hay The maximum charge, being determined by wz, will be the same with alkali as with acid. Of course, the influence of the oppositely charged ion in the acid or ie base is being neglected. Sa CasE 2.—The surface is not chemically inert : preferential | adsorption of one ion 1s possible. A review of the literature shows that surfaces in contact with water are seldom neutral. They are generally more or: less negatively charged. This is intelligible in view of the the Adsorption of Lons. 335 chemical! reactivity of the hydroxyl group. The presence of the potentially tetravalent oxygen atom possibly leads to a selective adsorption of hydroxyl i ions by most surfaces. Thus glass and quartz have'a “marked negative charge in contact with water (cp. Elissafoff). On the addition of an acid the electrostatic forces will produce a diminution of the charge. ‘The electrical adsorption of hydrogen ions by hydroxyl ions cannot be distinguished from the recombination of hydrogen and hydroxy! ions to form neutral molecules of water. This is confirmed by the fact that the equation of electrical ad- sorption (cp. previous paper) satisfactorily represents the diminution of the charge. Perrin (loc. cié) found that, excepting alumina and chromium chloride, all other substances (naphthalene, silver chloride, boric acid, sulphur, salol, carborundum, gelatine, and cellulose) show a preferential adsorption of hydroxy! ions. The sur- faces have a negative charge even in contact with acid solutions. He also found that at higher concentrations of the acid the surface acquired a positive charge. LElissafoff, McTaggart, Ellis, Powis, and others could not observe this reversal in their investigations. Hlectrical adsorption of hydrogen ions cannot lead to a reversal of the charge. The reversal (or the non-reversal) of the charge becomes intelli- gible if it is assumed that the considerations set forth in deducing equations (8) or (9) are correct. In contact with pure water the surface has a layer of adsorbed water and a number of hydroxyl ions. The amount of hydroxyl ions adsorbed by the surface will, in general, be small, as the concentration of the hydroxyl lons is very small in pure water. If, however, the adsorption is very strong the surface will have a considerable negative charge. On the addition of an alkali the negative charge of the surface will increase, due to two reasons: (1) the preferential adsorption of hydroxyl ions will Bec. and (2) the number of hydrogen ions being formed at the surface will be more and more neutralized by hydroxyl ions in the liquid (cp. scheme 2(a)). A maximum will be reached when the surface is saturated by preferential ad- sorption and when @=1 in equation (9). The maximum charge per unit area can be written as ewe ie (Lor mean, 2) CRY where “wz” corresponds to the charge when 6=1 in equation (8) and ‘y” is proportional to the number of | ; | OEE Doo Prof. J. N. Mukherjee on hydroxyl ions the surface can adsorb per unit area when it is saturated. Since the chemical adsorption of hydrogen ions is assumed to be absent, on the addition of an acid the; negative charge will decrease owing to electrical adsorption till the surface becomes neutral. At this concentration ot the acid, the surface has an adsorbed layer of water, and an equal number of hydrogen and hydroxyl ions. An increase in the positive charge eannot be due to electrical adsorption of the univalent hydrogen ions (cp. previous paper). The increase in the charge is due to the neutralization of the hydroxyl ions being formed in the surface by impinging hydrogen ions, as represented in scheme 2 (6) above. The maximum charge E,, for an acid will, therefore, be equal to “wz.” The maximum charge due to acids thus gives a measure of the hydration of the surface. The difference between the maximum charge observed with acid and with | alkali gives a measure of the amount of hydroxyl ions that is required to saturate the surface. In tne preceding discussion, the chemical and electrical adsorption of the anion of the acid has been left out of account for the sake of simplicity. If the initial negative charge of the surface in contact with pure water is consider- able the electrical adsorption can be complete only at high concentrations of the acid, 2. e., the surface will be neutral at a high concentration of the acid. The electrical adsorption of the anionis no longer negligible. A reversal of the charge, though theoretically possible, may not be actually observed owing to the great concentration of the anion. The reversal is thus dependent on :— (1) a large value of w, and (2) a small value of y. A non-reyersal is to be expected when the opposite is the ease, 7. é., (1) a small value of *w,” and (2) a large value of “y.” A regular transition from marked reversal to non-reversal can be observed in Perrin’s work. With cellulose he also does not record a reversal of the charge. It is to be expected from the preceding considerations that non-reversal will not be observed when the concentration of the acid required to render the surface neutral is comparatively high, 7. ¢., the anion concentration is high. The concentration of the acid in the case of cellulose is the greatest recorded by Perrin. the Adsorption of Ions. d3% The chemical adsorption of the anion is also not to be neglected. The experimental data on this subject are meagre. The various points raised here can be experimentally eluci- dated. As shown above, the standpoint developed in this paper can correlate all the observed facts. Besides, it gives a definite idea of the electrical double jayer. Adsorption of electrolytes. In the preceding sections the adsorption of ions has been considered with reference to the electrical charge of surfaces in contact with aqueous solutions of a single electrolyte. The electric effects accompanying the adsorption of ions have enabled us to follow the total adsorption of ions of both signs. In considering the adsorption of ions measured by chemical means it is important to remember the influence of the ad- sorption of the solvent pointed out by Arrhenius, Bancroft, Williams, and others. The amount adsorbed is small and the analytical measure- ment is difficult. For this reason, investigations have centred round adsorbents with great adsorbing power and substances which are strongly adsorbed. Often it happens, that if a sub- stance is used in a satisfactorily pure state it does not have the necessary specific surface to make the estimation of the adsorbed amount possible. Asa result adsorbents generally contain small amounts of other substances. The importance of these impurities has been pointed out by some investigators. Michaelis and Freundlich and their co-workers have done systematic work in this field. Their investigations have brought out the following regularities :— | (a) The electric charge of the solid influences the ad- sorption, Thus Michaelis and Lachs (Z. Hlektro-Chem. xvii. poe vi CITE); Biochem Zeresch. xxv ap. 309. (L910)... and Davidsohn, Biochem. Zeitsch. liv. p. 323 (1913)) found that in contact with acid solutions charcoal adsorbs anions strongly and does not adsorb cations. The reverse happens in the case of cations. Freundlich and Poser (Koil. Chem. Beihefte, vi. p..297 (1914) ) undertook an extensive investiga- tion, and they agree with Michaelis as to the electro-chemical nature of the adsorption. ; (b) The chemical nature of the adsorbent has a specific action. Michaelis and Rona (biochem. Zeitsch. xevii. pp. 57, 85 (1919)) believe that adsorption is due to chemical affinity. They mention that charcoal has a great capacity for adsorbing substances containing a chain of carbon atoms. (Cp. Abder- halden and Fodor, Fermentforschung, ii. p. 74 (1917).) Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. Z 338 Prof. J. N. Mukherjee on Freundlich and Poser (loc. cit.) found that the nature of the adsorbent plays an important part in determining the ad- sorbability of a dye. Both Michaelis and Freundlich agree that at least two types of adsorption of ions can be recognized. (c). Exchange or displacement of ions alr eady adsorbed by ions of a second electrolyte (cp. Freundlich, “ Verdriingende Tonenadsorption ” and Michaelis, “ Austausch- Adsorption ”’). Michaelis (Z. Electrochem. xiv. p. 353 (1918) ) considers that a substance like mastic, or kaolin (bolus), acts as a “‘ zweler electrode’’ (a binary electrode). Thus kaolin has a slow- moving anion (silicate ion) anchored on its surface and tends to send hydrogen ions into the solution under a definite electrolytic solution tension. Freundlich points out (and Poser, oc. cit.) that other cations can displace the hydrogen ions and form undissociated complexes (and HElissafoff, Z. Phys: (Chem 0: 33) 1912). (d) An adsorbent which contains some adsorbed electro- lytes need not be necessarily saturated. In this case, besides an exchange of ions, primary adsorption of ions is possible. This also applies to substances which act as binary electrodes in the sense in which the word has been used by Michaelis. He considers that, besides adsorption through exchange of ions, there is only one other type of adsorption, namely, adsorption of both ions in equivalent amounts (“ Aquivalent Adsorption ”’). One other fact has been emphasized ie these authors. (ec) It is the irreversible nature of electro-chemical ad- sorption. The well known instance of the adsorption of hydrogen sulphide by metal sulphides studied by Linder and Picton (T. Ixvii. p. 163 (1895) ; Whitney and Ober, J. Amer. Chem. Soc. xxii. p. 842 (1901)) can be mentioned. The adsorbed substance does not come out in solution when the adsorbent is brought in contact with pure water. (f) Lastly there is no clearly established instance in which Rael alte splitting up of neutral salts such as potassium chloride has been observed through adsorption. Theories regarding the Hachange of Ions. The conception of an adsorbent acting as a binary electrode, suggested by Michaelis, is not of much help in explaining the exchange of ions and other peculiarities of the adsorption of electrolytes. The relationship between the adsorption of ions, electro-endosmotic cataphoresis, and precipitation of colloids has been established beyond doubt. The only theory — . the Adsorption of Ions. o0U that attempts to correlate them is that due to Freundlich. This view is an extension of Michaelis’s idea referred to above. The adsorbent (or colloidal particle) is regarded as a great multivalent ion (cp. Billiter, Z. Phys. Chem. xlv. p. 307 (1903) 3 Duclaux, J. Cham. Phis. vi p: 29 (1907)). . The following extract shows clearly their standpoint (Freundlich and HElissatoff, loc. cit. p. 411) :— “Die Ladung soll nun durch die verschieden grosse Lidsungstension der lonen des schwerldslichen festen Stofts zustande kommen, aus dem das suspendierte Teilchen, bzw. die Wand besteht. Nimmt man als Beispiel das Glas, so hat man an der Oberfliiche desselben eine Schicht von gelostem, oder bzw. in wasser gequollenem Silkat; die K- and Na- Ionen haben eine grosse Lisungstension und bilden eine ftussere Schicht, die ‘schwerldslichen, langsam diffundierenden (vielleicht auch stark absorbierbaren) Silikationen bilden eine innere Schicht, die mit dem festen Stoff verbunden wie ein vielwertiges Ton sich verhalt. Der wesentliche Unter- schied gegen ein gewohnliches Ion liext darin, dass wegen der Groésse Grenafliichenwirkungen eintreten, die Konzentra- tion ist in der Umgebung dpe so homogen, sondern es sind durch Adsorption hervorgerufene Konzentrationsunterschiede vornanden. “Fiir zwei ionen gilt nach der Massenwirkungs gesetzat (Anion) . (Kation) = i (nichtdissociertes Salz), deshalb auch fiir das aaa vielwertige Anion des als Beispiel betrachteten Glases. ‘(Vielwertiges Anion) . (Kation) = K (nichtdissocierter Stoff). Es wird also von der Konzentration der Kationen die ‘ Konzentration des vielwertigen Anions,’ d.h. auch die Zahl der auf der Grenzfliche vorhandenen Ladungen abhingen. “ Die Kationenkonzentration, um die es sich hier handelt, wird aber in erster Linie die dee nichsten Umgebung der Grenzfliche, d.h. die Adsorptionschicht sein. Die adsorbierte Menge Kation wird also fur die ‘ Konzentration des viel- wertigen Anions’ d.h. fiir die Ladung der Grensfliiche mass- eebend sein. Dies ist eine andere Verkniipfung von Adsorption und Potentialdifferenz an der Grenzfliiche. Genau das Gleiche gilt naturlich fiir ein vielwertiges Kation und die adsorbierten Anionen.” There are several difficulties in accepting this theory. Salts of alkali metals can neutralize. charged surfaces at moderate concentrations (N/10 or N/20). One has to con- clude that the alkali salts of these ‘* multivalent anions” have a low solubility product. The effect of the valency of the ZL 2 340 Prof. J. N. Mukherjee on oppositely charged ion cannot be accounted for. The activity of the cations is generally in the following order :— fh > Al > Ba > Sr Cae > Cs > Rb > he Nae The postulates that alkali metal salts become undissociated at low concentration of the cation and that their solubility products are of the above order for a large number of diverse chemical snbstances, are contrary to experience. Regarded from the chemical point of view the generality of these ob- servations cannot be explained. Besides, the conception of the suppression of the dissociation of a salt cannot explain the reversal of the charge which is met with when the oppositely charged ion is polyvalent. The view of electrical adsorption put forth by the writer gives a definite correlated account of these various facts. The Role of Llectrostatic Forces in the Absorption of Tons. (a) Adsorbent in contact with a single electrolyte :— Let us consider an adsorbent, P,in contact with an electro- lyte A’ B. It is assumed that the substance P only adsorbs the anion B’ by chemical affinity. For simplicity it is also assumed that “‘P”’ is a pure chemical substance of definite composition. The amount of B- adsorbed per unit area will depend on the concentration of A* B™ and on the streneth of the chemical affinities acting on B . Corresponding to the number of anions adsorbed an equivalent number of cations A” remain in the solution. These are held near the surface by electrostatic forces, and form the second mobile sheet of the double layer (cp. the earlier paper referred to). If the concentration of the electrolyte is sufficient, some of them will be fixed on the surface by electrostatic forces. These ions of opposite charge fixed on the surface by electro- static forces will be spoken of as electrically adsorbed in the sequel. The chemical adsorption of an ion thus concen- trates both ions at the surface in equal amounts. That is, the primary adsorption is an equivalent adsorption of both ions. Analytical methods cannot differentiate between the two adsorptions, but electro-osmotic and cataphoretic experi- ments can (cp. (d) above). If the adsorption of the anion is due to strong chemical forces, perceptible amounts of the electrolyte A*B°™ will be adsorbed at very low concentrations. Hven saturation may be reached at low concentrations. In such cases, if the the Adsorption of Lons. 341 adsorbent with adsorbed electrolyte is suspended in pure water the adsorbed electrolyte will not be set free (e). Since the primary adsorption of the ions is due to chemical affinity, the influences of the nature of the adsorbent and of the electrolyte (b) are intelligible. (6) The addition of a second electrolyte :— The general case when both electrolytes, A” B™ and OT D7, are present in all possible concentrations will be too complex. It will be assumed for the sake of simplicity that (1) the substance P adsorbs chemically the anion B™ strongly, and that the concentration of the electrolyte A’ B™ in the liquid is negligible. We are thus dealing with an adsorbent with an amount of adsorbed electrolyte in contact with a second electrolyte solution ; (2) the atoms on the surface of the adsorbent P do not exert any chemical affinity on the ions C* and D™. This particular case corresponds with most actual systems, and the electrolyte A‘ B” plays the part of the “ Aktiver Hlectrolyt” of Michaelis. Let us now consider the effects of the electrostatic forces on the ions OC’ and D.. A cation ©", when it diffuses into the double layer owing to thermal energy, will be attracted to the surface. Considering the kinetic equilibrium between the ions in the second sheet of the doyble layer (Ae and C") and those in the liquid, it is evident that the relative propor- tion of A‘ and (‘" ions in the double layer will depend on (i.) their respective concentrations in the bulk of the liquid, and (i1.) their valency. The same consideration applies to the electricaily-adsorbed ions A* or O°. At. sufficiently large concentrations the whole of the mobile second layer and electrically-adsorbed ions will be formed by the ions C*. There will thus be an exchange of ions, and the amount of exchange will depend on the concentration of the second electrolyte. When the displacement is complete the amounts exchanged will be equivalent to the amount of B- ions primarily adsorbed and independent of the nature of the replacing ion C’—a fact often observed (cp. (Linder and Picton, loc. cit.; Whitney and Ober, loc. cit. etc.). The ions CG will be positively adsorbed. The relationship between the charge of the surface and the positive adsorption of the oppositely-charged ion is also obvious. The amount of (” ions absorbed depends on the 342 Prof. J. N. Mukherjee on amount of the negative ions chemically adsorbed. es that increases he total amount of adsorbed negative ions will increase the positive adsorption of cr. The reverse case, when the positively-charged ions A” are adsorbed chemically instead of the ions B’, and no other ions are chemically acted on by the surface, is obvious. Negative ions will now be positively adsorbed and exchanged. This state of affairs corresponds with the statements made in (a) and (c) above. Taking the same case again, we shall consider the effect of the electrostatic forces on the ion D - Anion D , diffusing into the double layer, will be driven out of it. So long as the potential of the double layer is sufficiently strong, a volume of the liquid equal to “S/,” where ‘‘S ” is the extent of the surface and “1” is the thickness of the double layer— will be free from the ion D . In other words, the concen- tration of D” increases in the bulk of the liquid and a negative adsorption will takeplace. This will increase with the concentration of the electrolyte so long as the potential of the double layer is sufficiently strong. Since with in- crease in concentration the potential falls, the negative adsorption will reach a maximum. At concentrations when the surface becomes electrically neutral, there should be no negative adsorption due to electric forces. It is difficult to determine negative adsorption at high concentrations as the osmotic pressure opposes it. Also, the variations in concen- tration due to negative adsorption become relatively small. The experimental ‘difficulties lie in the analytical estimation of small amounts. Only ions which ean be estimated in extremely small amounts are suitable for experiment. Hstrup, (Moll. Zeitsch. xi. p. 3 (1911)) thas Vacnuallig observed a negative adsorption of the oppositely-charged ion. He estimated the adsorption of the iodate, dichromate, and chromate of ammonium. Michaelis and Lachs (Koll. Zeitsch. ix. p. 275 (1911)) did not observe a negative -adsorption with potassium chloride. Eixactly similar observations have recently been made by Bethe (Wiener Mediz. Wochsch. 1916, Nr. 14; Koll. Zeitsch. xxi. p. 47 (1917)). He worked with gelatine gel, gelatine sol, and a number of animal cells. The adsorption of a basie dye is greater in weak alkaline solutions tae in neutral solutions. The same is the case for an.acidic dye in weak acid solutions. In alkaline solutions the adsorption of acid dyes is negative, and the same is the case with basic dyes in acid solutions. Examples of the role of: the electrical the Adsorption of Ions. 343 force in the adsorption of ions can be multiplied (cp. Baur, Z. Phys. Chem. xcii. p. 81 (1916) ; Michaelis and Davidsohn, foc. Cit.) ; Exchange of Bases in Soil and Soil-Acidity It is now easy to understand the nature of the exchange of bases in soil-analysis and the cause of soil-acidity. A com- plete reference to the older literature is given in the following papers :— (1) McCall, Hildebrandt, and Johnson, J. Phys. Chem. $9265 SK.\ps O1. (2) Ruce, ibid. p. 214; (3) Truog, ibid. p. 457. Russell (Brit. Assoc. Rep. 1918, p. 70) has given an excellent summary of the present position of the subject. The facts are that— (a) Neutral solutions of salts like potassium chloride, if treated with samples of soil, give acid extracts though the extract with pure water is neutral, (>) In a large number of cases it has been shown that there is a definite exchange of the cations. Equivalent amounts of bases are exchanged in many cases. | Two different views have been advanced to explain the | facts. The older chemical view regards the process as a chemical interaction between definite acids (e. g., humus | : acid) or complex salts (e.g., silicates) and salt solutions. The other view begins with von Bemmelen, and regards it | as an adsorption process. Cameron suggested (cp. Russell’s | Report) that the soil adsorbs the base more strongly than it | adsorbs the acid. ; The objections against the chemical view can be sum- | marized as follows:—The extract with pure water being ) neutral, the soil-acids must be insoluble. ‘The acids must be : unusually strong, as they evideutly decompose a neutral | salt solution combining with the base, liberating the strongest known acids, like hydrochloric acid. Evidently such acids are unknown, and it is difficult to conceive of such reactions. Regarding the exchange of bases, the difficulty lies in the assumption that the basic ion is taken up to form an insoluble salt. It is necessary to postulate the existence of insoluble salts of alkali metals in a : large number of cases (cp. the remarks on Freundlich’s ' theory). : That adsorption plays an important part is also evident from the works of Russell and Prescott (J. Agric. Sci. vill. p. 65 (1916)) on the interaction of dilute acids and phos- phates present in the soil. But the view of Cameron does o44 Prof. J. N. Mukherjee on not seem to be tenable. The preferential adsorption of an ion by the soil does not mean hydrolytic decomposition of the salt. It appears from the summary given by Russell that the equivalent exchange of bases lies in the way of regarding the reaction as an adsorption process (loc. cit. pp. 71, 75, 76). It would be apparent from the previous discussion that this, in itself, does not contradict the adsorp- tion hypothesis. Soil can be regarded as a complex colloidal system. It is a complex gel consisting of aluminium and other silicates, free silica, ferric hydroxide, ete. The gel is mixed with insoluble crystalloids. It also contains small quantities of adsorbed electrolytes and organic matter in indefinite and varying proportions. The gel adsorbs anions by chemical affinity. These anions may be :— (1) of organic acids, such as humus acid ; (2) of simple electrolytes like chlorides, sulphates, car- bonates, ete. ; (3) hydroxyl] ions rom ater, Owing to the complex chemical nature of the gel and the enormous specific surface of gels, large quantities of anions may be adsorbed. An equivalent number of cations remain near the surface as the mobile second sheet or as electrically adsorbed. The exchange of bases is simply due to the dis- placement of these ions. When the displacementis quantitative equivalent amounts are exchanged. ‘The anions primarily adsorbed or the cations in the second sheet are not of one kind. The relative numbers and chemical natures of these ions will evidently vary with the different soils. An extract with pure water will be neutral unless the soil contains free acids. An extract with a neutral salt can only be acid when the cations displaced from the second sheet (or electrically adsorbed) contain hydrogen ions or such ions as aluminium, which~ hydrolyse in dilute aqueous ‘solutions. The role of the aluminium ions in determining the acidity of the soil extract has been pointed out ioe Daikuhara (Bull. Imp. Central Agri. Expt. Station, Tokio, ii. pp. 1-40 (1914)), and has been fully confirmed by Rice (Loe. cit.). The function of organic acids has constituted a oreat objection against the adsorption hypothesis. The hydrogen ions in the second sheet have probably, in most cases, their origin in these acids. This view thus correlates the exchange of bases observed with soil with such exchanges as have been observed in the adsorption of electrolytes (ep. Michaelis). That sometimes considerable quantities of bases are ex- changed should be referred to the enormous surface of these the Adsorption of Lons. 345 gels, and that probably the surface is saturated with anions. As erystalloids (insoluble) are also present, the type of exchange considered by Paneth (loc. cit.) is also possible. It is needless to point out that in this discussion only the theoretically simple case has been considered. Complications due to simultaneous primary adsorption of different ions and their mutual displacement are not always negligible. Besides, the changes may not be restricted to the surface ; for faiion of solid ‘solutions, etc., are not excluded. Considering all these complex influences, it is interesting to note that most ot the observed regularities correspond to the theoretically simple case. Adsorption of Ions in its Relation to Permeability of Membranes and to Negative Osmosis. In conclusion, a few remarks will be made on the funda- mental interest that a study of the adsorption of ions has for biological phenomena. Cell activity 1 is greatly conditioned by the permeability of its ‘‘ walls” or the cell-substance to the contents of the liquid with which it is in contact. The connexion between the rate of osmotic flow through mem- branes and even the direction of the flow, and the potential differences existing on the two sides of the membranes, has been clearly established (Girard, C. &. cxlvi. p. 927 (1908), and following authors: Bartell, J. Amer. Chem. Soc. Xxxvl. p. 646 (1914) ; Hamburger, Z. Phys. Chem. xcii. p- 885 (1917)). The origin of the potential difference is generally assumed to be due to the fact that the rate of diffusion of the electr olytic 1 ions in the membrane substances is different from that in water. That the membrane potential is due to a selective permeability of ions was first suggested by Ostwald (7. Phys. Chem. vi. p. 71 (1890)) ; Donnan (Z. Elektrochem. xvii. p. 572 (1911)) has discussed the origin of the potential differences theoretically, and has given it a quantitative form based on thermodynamic considerations. In collaboration with others he has carried out a number of investigations which have established the validity of this view. The simpler case of a potential difference between two interfaces when an immiscible liquid is placed between two aqueous solutions has also attracted a good deal of attention. The work of Loeb and bis co-workers on cell- permeability and origin of the membrane potential is of fundamental importance (Loch and Beutner, Biochem. Zeit. li. p. 295 (1913) ; Beutner, 7. Phys. Chem. |xxxvil. p. 385 346 Prof. W. M. Hicks on certain Assumptions wn the (1914), 2. Hlektrochem. xix. pp. 329)/473 (1913); Loep J. Gen. Phys. xx. po 113 (1909). a1.) i273, 290510 eee 563, 673, 659). The part played by the adsorption of ions in these phenomena is twofold. The origin of the potential is In many instances due to the adsorption of ions (cp. Baur, Z. Elektrochem. xix. p. 590 (19138); Z. Phys. Chem. xcii- PONG co Sly Secondly, the electrostatic forces of the surface probably determine the relative permeabilities of the two ions. To this the semi-permeability of an ion can be referred. eee negative osmosis, attention may be drawn to the suggestion of Freundlich (Koll. Zeitsch. xviii. p. 1 (1916)) that the thin walls of the membrane substance conduct electricity, and electro-osmotic flow of the liquid occurs. A necessary condition is that one ion is permeable and the other relatively impermeable. This explanation meets thermodynamical requirements, and is the only satis- factory one hitherto put forward. In all these cases the same influences of polyvalent ions and ions of opposite charge are noticeable. The change in the collodial properties of the membrane is an important additional factor which has to be remembered. The influence of the electrostatic forces is unmistakable. Physical Chemistry Department, University College, London. XXX. On certain Assumptions in the Quantum-Orbit Theory of Spectra. By W. M. Hicks, &R.S.* (ee practically complete success of the quantum-orbit theory in describing all the known facts of spectra, in cases where we know experimentally that the source consists of a single nucleus and a single electron, must give assurance that the same procedure must also be capable of application to more complicated atoms than those of the hydrogen and enhanced helium types. Unfortunately, however, mathematical difficulties have so far prevented any rigorous application of the theory to definite cases, even of the next simplest atomic configuration of a single nucleus and two electrons. The attempt of Sommerfeld at an approximate solution shows, on the one hand, how hopeful we may be of a description of spectra on this ‘basis, and at the same time how far we are at present from its * Communicated by the Author. Quantum-Orbit Theory of Spectra. “B47 achievement. In the present. note I wish to illustrate this by drawing attention to certain assumptions as to actual spectral data, which have been made and which do not appear to be justified. The criticisms may not affect essential points, but they would appear to require some modification in the presentment of the theory. References will be made to Sommerfeld’s ‘Atombau und Spektrallinien,’ 2nd edition (1921). L. Sommerfeld (pp. 276, 506) takes a configuration of a central nucleus, surrounded by a ring of equally-spaced electrons, and at a considerable distance furiher out one electron revolving in a quantized orbit. On the assumption —here justified—that the ring can be treated as if the whole charge of the electrons on it were continuously distributed along it, he obtains as an approximation the same form for a sagan function (or term) p as that’ suggested by re VIZ. = N/{m+ptap}?. He says that silva, is the actual true form, as already deter- mined by observation. This is, pee by no means the ease. No form has yet been found which will fit in for all series, and indeed the form N/(m+p+a/m)* is in general rather superior to that of Ritz. It is to be noted that the assumption made above leads to the same result ag if the force to the centre depended only on forces inyersely as even powers of the distance, and forces depending on odd powers—say 1/r*—are excluded. It may also be noted in passing that the theory so developed applies only to the case of a single external electron and one internal ring, that is, on the usually assumed configuration of eight- electron rings, only to the spectra of the fluorine group, or the ionized rare gases, or the doubly-ionized alkalies, ete. By taking his E as (k—s;’e in place of ke, the formula would meet the more general case. This modification, however, would only slightly affect the order of magnitude of the quantities pw, a. In the formula m=n-+n’', where n, n' are respectively azimuthal and radial quantum numbers, and yp, e& are functions of n and not of n’. * Asa result of successive approximation, « being small,:this means for a complete approximation the form pH=N/{m+ pta/(m+p)?+B/(mtp)i+...P, which, «8 is well known, is capable of reproducing practically all cases if p,a, 8... are all at disposal, and are not related necessarily to one another, as here. 348 Prof. W. M. Hicks on certain Assumptions in the It is not to be expected that the numerical values of the constants 4, « on this special theory should accord with any determined by experiment, but they should be of a suitable order of magnitude and general character. It may be interesting to test this. The expressions for the constants: fs, a may be written > 3 k—s, p= Ey i4 PS (1-3 St) 92 boa n 4. DL— where (277)4m?e? E(Z— k) 5 (3 = A} : = 8°9(Z—k)(k—s;)10!%r? and ine 2° k—se If p be measured in wave number instead of frequency, the a must be multiplied by the velocity of light. Then ee Ope ee eae) 1010p? ee ‘i n? Zi k— sp, : Here v denotes the radius of the internal ring in em., Z the atomic number of the element, k the number of external electrons, and s;, depends on the mutual action of the external electrons on‘one of them. For Li, Z=11, k=3, 5% =:577, and a 37r?m € L— k ) \ ee he Deh on 9: ; e ms =-(1 +22 10048) 1018, ele as 13 10192 Voinn (ihe In actual cases, for wave numbers of about p=10°, pac/p lies between about -9 and ‘01. Hence the second equation requires r<10-7°>10-*°. Since «<1, the first equation requires 7 to be about 10-®, but as the second term in the bracket is determined by an approximation it must be a small fraction, whence r<10-8 >. The fact that both give values of the same order of magnitude, even if they cannot. exactly agree, and not far from what might be expected for an 8-electron ring, is certainly satisfactory. 2. It is deduced that the different types of sequences correspond to azimuthal numbers n=1, 2, 3, 4, the different orders of the same type to radial quanta n'=0, 1,.... These are then co-ordinated with the s, p, d, f types because it is stated that these types have their lowest orders respectively of 1, 2,3, 4. It is difficult to see how this statement has been arrived at, as it is quite incorrect. For the sake of Quantum-Orbit Theory of Spectra. 349 readers who may not be familiar with spectral data, it may be well to consider them here. (s, p.) For the s, p the lowest order are: s p. Seal (ps Rare gases ......... 1 1 II { Alkaline earths... 2 1 (The alkalies ...... l 2 * ais Cabin, bho. (ae il eta i Ane. [ib Ave Mer cet eimany 1 it ARR Bray If it were not for the cases of the rare gases and the Cu subgroup, the assumption might be explained by an interchange of the nature of the sequences which produce P, S series (for which in Groups I. and IJ. indeed there is also direct evidence). But that two groups make s, p both have unity for their first order is fatal. (d.) The assumption of 3 as the first order for d(m} no doubt is based on the fact that Ritz made it in dealing with the D series in the alkalies. The denominators of the first orders in this group are comparable with 2°9, which Ritz wrote as 3—‘1 and called the first order 3. But this procedure is inadmissible either on the side of the formula or from what we know of the constitution of the d sequence. In Sommerfeld’s formula w is positive, and it is only by treating the fraction as positive that we find a detinite dependence of it on certain spectral constants. But even so, the first order is not 2 for all groups. The law of the first order of the d sequence is a quite simple and definite one, and is given on p. 188 of my recently published ‘ Analysis of Spectra. It is that in each group of the periodic series, the subgroup of elements whose melting-points increase with atomic weight take as their first order m=1, whilst the sub- group with decreasing melting-points take m=2. (f.) In the case of / the 3+fraction has again clearly been written 4—/, and the assumption has been made that the lowest orders of the f type take m=4. But here also, for the same reasons as in d, the fraction must be taken as positive. In the alkalies certainly the lowest order observed is F(3), but F(2) would lie far up in the ultra-red, beyond even Paschen’s longest lines. In the Cu subgroup there is evidence for m=2 and indications for m=1. The alkaline earths have m=2 both in triplets and doublets. In the Zn subgroup only F(3) has been observed, but F(2) would lie in the extreme ultra-red. In Group III. there is no evidence, whilst in the rare gases there are examples of F(1) and F(2). | It would thus appear that the theoretical deduction that different types depend on successive changes of azimuthal quanta by unity is not tenable. 350 Prof. W. M. Hicks on certain Assumptions in the 3. In dealing with the Zeeman effect on p. 422, Sommer- feld adopts Puaschen and Back’s interpretation of their experiments on the Zeeman effect in the case of close multiple lines.. This interpretation was based on precon- ceptions as to the nature of the series types in He and Li, which they investigated. I have given * reasons why this interpretation should be modified. On either inter- pretation, however, a consequence follows which appears difficult to explain on the quantum-orbit theory. Take, for example, the case of the helium doublet at 4713 A. Hach cemponent in weak fields shows special Zeeman patterns. Wath increasing fields and consequent approximation of certain constituents from each pattern, an interaction occurs of one on the other. Such an effect can only be produced if the two patterns are produced in the same source. Hence the original components of such a doublet must be produced simultaneously in each atomic configuration, whether a magnetic field is present or not. It follows that in radiation . there must be simultaneous passages of two electrons, each from its original orbit to its final one. But as the effect takes place at one operation, the total change of energy is passed on to the radiator and emitted as a single mono- chromatic radiation, 7. e. no doublet. It might be suggested that the effect could be explained on the hy pothesis that the magnetic field affects the mutual possible orbits, and that sometimes one passage occurs and sometimes the other. It is difficult to see, however, how an orbit can be modified by another supposed one which is non-existent, 7. e. not being described at the same time. 4. This consideration does not affect evidence for the quantum theory, but will serve to illustrate a habit which is somewhat exasperating in reading the writings of many exponents of the quantum theory—viz., the picking up of small and often irrelevant points as charming results of the theory. On p. 300 ff. it 1s expected that each doublet separation on passage from arc to spark conditions should be magnified in a measure corresponding to the ratio 4N:N, and satisfaction is expressed that in data adduced from corresponding elements in the doublets of group I. and the enhanced doublets of group JI. this expectation is fulfilled. The ratios of the separations are reproduced (with Hg: Au added) in the first line of the following : Mg. Ca. Sr. IB) tae PAK Cd. Eu. Hg. 53 3°9 ot Bll aS) ir ? 2°5 2°24 2:10 205 1:92 Deal 2°12 2°03 2:12 * ‘Analysis of Spectra,’ § 7, p. 96. Quantum-Orlit Theory of Spectra. oer But surely these numbers show that the comparison is not justified. As is known, a correspondence actually lies between the enhanced doublets and the triplets in the same element. Thus the ratios of the doublets to the first separation of the triplets are given in the second line of figures above, where the agreement is remarkably close. In this latter case, however, the correlation is not direct. It is due to three foncurrent facts: (1) ratio 4N:N; (2) the oun multiples in the doublets and vy, of the triplets are very nearly the same in each element ; and (3) the denominators in the doublets and triplets have nearly the same ratio in all (see below). ‘There is, however, a close correspondence between the mantisse of the doublets in the two groups I. and II., especially as between the alkalies and the alkaline earths, fleas of the latter being about double the first. Correlation is also shown between the denominators of the triplet and doublet sets in all the group IL. elements. These statements are illustrated by the following data :— IL. i ee Tae aot mri too Tripl. Doubi. Ratio. Doubl. Ratio. 1 ee 1-660 2:265 1:36 DN) ha 2117 aya) a es Poo) 249 peat Wy Biss. 4. 1, 2-935 9-12 Sh eee 1:880 2611 1034 aaa EO etc 2-292 2:09 eT Hs os 1:957 2139 1:39 Ca. ee 2°361 2°03 ee 1°5$9 2-098 1S ie | Ck ee 1:869 he 1641 2144 1°30 ve tee 1:892 Ho’ fe. 1:648 9-190 132 vas 5S ba teats 1°653 £ — ANU aaa 1-929 Here under Ll. the third column gives the ratio of the denominators ; under I. the second column gives the ratio o7 mantissie in LI, to those in I. On the other hand, there appears very little correlation between the oun multiples which give the separations in corresponding elements of groups I. and Il. A much closer one is found between those of the triplets and doublets of the same element in II. Thus, in the following, the first line gives the ratios of the oun multiples of doublets in group I. to those of the enhanced doublet of the corre- sponding element in group II. The second line gives the ratios of these multiples for the first triplet separation and the enhanced doublet in each element. Mg. Ca. Sr. Ba. Zu. Cd. Eu. He. 683 179 "843 903 ‘847 [71 ae ? “691 *709 "726 ‘761 ‘789 846 "842 ? B52.) XXXI. On the Theory of the Characteristic Curve of a Photo- graplic Emulsion. (Communication No. 22 from the British Photograpnic Research Association Laboratory.) ag. C.. oy MSc ainsi Ec Me nega seme li the most recent investigations on the relation between the photographic effect and the light-exposure, special plates containing only a single layer of grains have usually been employed. With such plates the photographic effect is determined by counting the percentage of grains made developabie. The curve expressing the relation between this percentage (wv) and the logarithm of the exposure may be called the characteristic curve of a single-layer emulsion, corresponding to the ordinary curve of a commercial emulsion, in which, instead of w, values of the density (in the photographic sense) are plotted. In a recent paper (Phot. Jour. 1921, Ixi. p. 417) the author has shown that such a curve, fora set of grains which are geometrically identical, is of ang usual S-shaped type, 2. e. a difference in size or shape does not account tor the fact that all the grains do not become developable with the same ex- posure. Now, a set of geometrically identical grains, all in a single layer and similarly orientated to the incident light, represents the simplest possible emulsion which we can in- vestigate experimentally. It also corresponds to the simplest theoretical case, eliminating many complicating factors which, though greatly affecting the form of the characteristic curve, have “nothing to do with the primary mechanism of the photograph process. In other words, with this emulsion the curve is reduced to its “‘ purest” form, and is determined almost solely by the photochemical process which takes lace. : It is now generally believed that the primary action of light on the grains is ce form in or on the surfaces of them certain “ centres” “»yoints of infection”? which act as starting-points for Hee reduction by the developer. This view has for some time had considerable evidence in its favour. Chapman Jones" (Phot. Jour W911, lr pai) showed that by stopping development at a very early stage it is possible to get particles of silver tco small to be visible microscopically, but which can be shown to be present by the colour imparted to the film, and by enlargement to visible dimensions by the deposition on them of mercury. Hodgson (Brit. Jour. Phot. 1917, p. 532) carried development a little * Communicated by Prof. A. W. Porter, F.R.S. Characteristic Curve of a Photographic Emulsion. 353 further, and showed it possible to observe the silver reduced by the developer only around certain centres in the grain, and a recent paper of Svedberg’s (Phot. Jour. April 1922) leaves little room for doubt that the possibility of a grain being made developable depends on the existence in it of some kind of reduction centre. Opinion as to the nature of these centres seems at present to be divided. There are those who assert that they are formed by the light-action, and that they do not exist before exposure is made. Such, for example, is the case if the centre is really a molecule of silver halide which has lost an electron, as is believed by H. 8. Allen (Phot. Journ. 1914, liv. p.175). On the other hand, there are those who believe that the centres are actual particles other than silver halide formed in the grains during precipitation and subsequent ripening, and that these only become susceptible to the action of developer after exposure to light. There certainly is considerable evidence to show thuit silver halide is not the only substance in the grains. Luppo- Cramer (Kolloidchemie und Photographie) was led, asa result of his work, to the conclusion that, at any rate in the most sensitive emulsions, nuclei are present which probably consist of a colloidal solution of silver in the halide. Renwick (J. S.C. I. 1920, xxxix. No. 12, 156T.) extends this idea, and says: ‘In our most highly sensitive photographic plates we are dealing with crystalline silver bromide in which, _ besides gelatin, some highly unstable form of colloidal silver exists in solid solution, and it is this dissolved silver which first undergoes change on exposure to light.’ These silver particles are negatively charged, and Renwick believes that the action of light is to discharge, and hence to coagulate into larger groups, those particles of colloidal silver which existed in the grain before exposure ; it is these groups of coagulated electrically neutral particles which are the re- duction centres. This view is supported by the ultra-micro- scopic observations of Galecki (Koll. Zeit. 1912, x. pp. 149- 150), who showed that X-rays have a coagulating effect on the particles in gold sols; by Svedberg (Koll. Zeit. 1909, iy. p. 238), who has similarly shown that ultra-violet light agglomerates ultra-microns to larger aggregates; by Spear, Jones, Neave, and Shlager (J. Amer. Chem. Soc. 1921, xliii. p. 1385), who have observed the same kind of effect with colloidal platinum; and by recent experiments of Weiger and Scholler (Sitz. Preuss. Akad. Wiss. Berlin, 1921, pp. 641-650). Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. 2 A ———————— —— 354 Mr. F. C. Toy on the Theory of the These facts are at any rate sufficient to justify an attempt to explain the relation of the number of grains changed to the light-intensity on the basis of the existence in the grains of actual particles which are not silver halide. These are not necessarily all changed to reduction centres from in- active particles by the same light-energy as they would be if they were single molecules of the halide. We shall make no assumptions as to the composition of these centres, and the theory does not depend on their being composed of colloidal silver. We shall use the term “nucleus” rather than centre to indicate the presence in the grains of actual particles before exposure. The Characteristic Curve of a Set of Geometrically Identical Grains. Theoretical. Our first object is to consider the case of a set of grains of identical size and shape, and to determine the relation we should expect to find between the percentage of these which are made developable and the light-intensity. The time of exposure is kept constant throughout. — If we consider a volume V of the silver halide which is very large compared with that of a single grain, we may assume that the total number of nuclei in any such volume of the emulsion is the same, though the number contained in individual grains in this volume may vary. We will define the sensitivity of a single nucleus as the minimum intensity which must be incident upon it in order to make it ‘ active ” in the presence of the developer. For a given intensity of the incident light there will be a definite number of such active nuclei in every volume V, and they will be distributed amongst the grains entirely haphazard, according to the laws of chance. Every gram which happens to have at least one active nucleus will be developable. When the intensity of the light increases, more grains are changed. On any “nucleus” “theory this happens because. more nuclei are present, so that a single grain has a greater chance of having at least one of them. This may be ex- plained in one of two ways. Firstly, all nuclei may have the same sensitivity, say I, but owing to the rapid absorption of light, those nuclei which are situated in the grain at some distance from the surface on which the light is incident, do © not receive an intensity of [ when the incident intensity is small. As the latter becomes greater, the volume of silver halide, throughout which the intensity is at least I, increases, Characteristic Curve of a Photographie Hmulsion. 355 so that the number of active nuclei increases also. Secondly, the sensitivity of every nucleus may not be the same, so that as the intensity of the light is increased, nuclei become operative which are unaffected by lower intensities, and again the total number of active nuclei increases with the intensity. We will consider only the case of grains in the form of thin plates as they oceur in high-speed emulsions. Hggert and Noddack (Preuss. Akad. Wiss. Berlin. Ber. 1921, xxxix. p- 631) have recently measured photometrically the fraction of the incident light which is absorbed by an ordinary commercial photographic plate, and have found it to vary with the different plates from about 4 to 12 per cent. for violet light, for which the amount of light absorbed is near the maximum. Now, these plates contain several layers of grains, so that a very extreme upper limit to the fraction of light absorbed by a single grain is, say, 20 per cent. hus, if there is an increase in the incident intensity of the order of 20 per cent., the intensity of the light transmitted through a grain will be equal to the intensity incident before the increase took place. Thus, if all nuclei are equally sensitive, a change in the incident intensity of the order of 20 per cent. will cause a difference in the number of active nuclei from zero to some fixed maximum, so that the characteristic curve can only function over a range of intensity such that the ratio of its extremes is of the order of 1°2:1. As will be shown later, for the steepest characteristic curve plotted this ratio is about 25 times as much as this, so that as an appreciable factor in determining the increase of nuclei with intensity the first assumption is untenable. We have, therefore, to assume that all the nuclei are not equally sensitive. Since these nuclei are all formed in the same emulsion, most of them will have a sensitivity near the average value for the whole, and there will be a few which are very sensitive and a few which are very insensitive. ‘There will be none which will respond to zero intensity, and none so in- sensitive that it takes an infinite intensity to affect them. We therefore expect the curve showing the relative number of nuclei R having any given sensitivity I to be of the general form shown in fig.1. The exact mathematical form of this curve is immaterial at present, but it will be similar in general form to that obtained by Clerk Maxwell for the distribution of velocities between the molecules of a gas. By similar reasoning to his, the number of nuclei (N,) which 2A 2 SS eee 356 ~ Mr. F.C. Toy on the Theory fo the have sensitivities between zero and I, (which is the number operative when the intensity of the light is I,) is given by Fig. 1. n—- e) the area OAB, 7. e. N=) /@Dd. on. a ee where f (1) gives the Ae of the ordinates in terms of I; or if No is the average number of nuclei per grain and a the number of grains in volume V, No= GQ) -al. oes The total number of nuclei is given by N= | Qa, t N, 0 es The curve showing the relation between No and I, shown in fig. 2, is characterized by its unsymmetrical-shaped § form, Characteristic Curve of a Photographic Emulsion. 357 When the a@ grains in the volume V of silver halide are subjected to an intensity 1,, every grain which happens to have at least one of these N, nuclei will be made developable. We have, therefore, to find the chance of a grain containing at least one of the N; nuclei when they are distributed hap- hazard amongst a grains. This can easily be obtained from the theory of probability. If p denotes the very small probability that an event will happen on a single trial, the probability P,. that it will happen r times in a very great number, say 7 trials, is (Mellor, ‘Higher Mathematics,’ p. 502) ee) omer a) eee 8) Let the volume of a single grain be v, then since the volume of every grain is the same the total volume V is av. Let p be the very small probability that a volume dv will contain a nucleus, then PN Geeta «Mme Sei al yal AL) To obtain the probability of the volume v containing a nucleus, we may suppose each dv to be a trial, so that the number of trials n is | eM eee |. menace 0") Ged) Therefore the value of np in equation (3) is N/a, which is equal to No. If in this number of trials the event (i.e. v containing a nucleus) happens once, a grain will contain one nucleus; if it happens r times it will contain 7 nuclei, so that from (3), (4), and (5) we see that the probability of a grain containing 7 nuclei is PES te irl) aL ughe (ae) which is the same equation as was obtained independently and first published by Svedberg. The probability of a grain containing ne nuelei is the value of this expression when e— 0, 2.2. : Paeo Now, since it is certain that a grain must contain either zero or at least one nucleus, the probability P, that a grain will have at least one is Pi be No, ° > ° . ° . ° (7) ew ee ee eee ~~ ] 358 Mr. F.C. Toy on the Theory of the But if w/a is the fraction of grains which are changed, Pi =aa ; ala=1—e%o, or, denoting log a/(a—a) by A, we have ANG Thus the same form of curve should be obtained when No is plotted against the intensity as is obtained when A is plotted against the same variable. The form of this curve should be an unsymmetrical 8, as shown in fig. 2. Haperimental. The first experiment carried out was to determine the relation between A and the light-intensity I for a set of geometrically identical grains, every grain counted being measured, as described in a previous paper (Phot. Jour. 192 xa. 41 e TABT EOI Cross-section of grain = 0°98’. Pane ce fou Be recs oe (dee values.) . | values.) 0 92°0 91:5 2°45 —0°895 | 47-7 466 0-62 —0:162 | 89°5 90-0 2°30 —1-215 52 52 0:05 —0°310 | 87°7 870 2°04 —1:487 0:0 0-0 0:00 —0°572 | 74:8 75:0 1°39 —1°788 0-0 0:0 0-00 In the first two columns of Table I. are shown the values of log I and # determined experimentally by exposure behind a neutral wedge, and these are plotted in fig. 3. The values of # given in column 3 are read off the curve in fig. 3, and it is these values which are used in calculating A in column 4. This is the best way of obtaining the A values, since when « is large a very small error in its determination means a very large error in A. The A, JI curve, shown by the solid line in fig. 4, is exactly as predicted by the theory. We must note here that this is not in agreement with the results of Slade and Higson (Proce. Roy. Soc. 1921, A, xeviii. p. 154) and a previous experiment of the author (ibid. 1921, c. p. 109) to confirm their result. Slade and Higson stated that the relation between A and I can be expressed by the equation Azal(1—e7*!), Characteristic Curve of a Photographic Emulsion. 359 where « and 8 are constants. A comparison of the form of this curve (fig. 5) and the curve in fig. 4 shows a difference Fig. 5. DOO) Ie” Mr. F. C. Toy on the Theory of the when I is large, but this can be explained. Firstly, the grains used in Slade and Higson’s experiments were not all of one size, the variation being about 30 times that in the present case. Secondly, the best curve given in Slade and Higson’s paper has actually the same form as that in fig. 4 if equal weight is given to each point plotted. Also, in the author’s contirmatory experiment, the main point was to show that at low intensities A varied, at any rate approximately, as I? ; at high intensities the work was not nearly as accurate Fig. 5. i Bl as in the present case. To be certain of the form of the curve in fig. 4 the upper part of it was plotted for another size of grain, and that the same result was obtained is shown by the dotted line. The position of a nucleus can be detected by Hodgson’s method of partial development of the exposed grains (ibid.). The developer used was made up as follows :— 200 ¢.c. saturated Na,SQsz, 8c¢.c. 10 per cent. KBr, 0°3 gm. Amidol. This is a weak, slow developer, and is best for this purpose because there is a bigger latitude than if the developer is strong in the time of development necessary to render the position of the nuclei visible and yet distinct from one another. The best development time was found by trial and examination of the grains under the microscope. After exposure the plate was plunged into the developer for a known time, then quickly and thoroughiy washed, and dried without fixing. The flat triangular grains used were so thin that. the silver deposit was visible without dissolving away Characteristic Curve of a Photographic Emulsion, 361 the silver bromide. In fig. 6 are given some examples of grains in which these nuclei appear ; they are formed more on the edges of the grains than anywhere else, though quite a number appear either inside or on the flat surfaces. Positions of nuclei numbered. The next experiment was to show if equation (6) holds good. A plate was exposed to a uniform intensity, partially developed, and the number of nuclei occuring on each of 150 grains was counted. Hence the average number per grain was known and also the number of grains having 0, 1, 2, 3, etc. nuclei each. In Table II. are given the theoretical and observed values of P,. for two equal-sized sets of grains, in one and the same emulsion, having widely different values of No. ‘TABLE EA: No=0°480. Se was | Value N®- grains Probability ‘Value No. grains Probability | | of 7. | camel Ounren. | Soe | moe Obs. Cale. 0 91 |0607 0619 | 0 43 |0-287 0-303. 1 47 10313 /0297 || 1 |° 55 {0367 0-362 | 2 | i |o073:\0071 | 2 | 36 [0-240 0-216 3 1 {0007 |0001 || 3 | 12 |0080 0-086 4 4 0-027 0-026 | 0 |0:000 - 0-001 The observed values of P, were determined by the fact that the probability of a grain having 7 nuclei is equal to the fraction obtained by dividing the number of grains which have r nuclei by the total number, 2.e.150. In fig. 7 the theoretical values are represented by the smooth curves, and ae SS Se Sa 362 Mr. F. C. Toy on the Theory of the those observed by the plotted points. The agreement is very good, and proves the validity of equation (6) in the case of a fast emulsion. : To find the relation between the average number of nuclei per grain and the intensity, a plate was exposed behind a step wedge and partially developed. The size of grain selected was the same as used for determining the A, I curve in fig. 4, the plate being exposed for approximately the same time behind the same wedge. At each intensity (1) the total number of nuclei on 200 grains was counted (except at I=0:044, where 100 grains were considered sufficient), and hence the average number per grain found. The values are given in Table ITI., and it will be seen that the curve in fig. 8 is of the same general form as the A,I curve in fig. 4 as is predicted by the theory. ? Tasue III. Cross-section of grain = 0°98,?. I. No. I No. 1-000 0:98 0270 | 0-24 0689 0°88 | 0-180 005 0490 0°63 0128 0:02 0356 0:43 0:044 0-00 Characteristic Curve of a Photographic Emulsion. 363 The highest value of No is about 1, which corresponds to less than 70 per cent. of grains changed, whereas actually the percentage changed corresponding to this value of Ny was about 90. This is because the partial development has not been sufficient to show up all the nuclei, and it is very difficult to do this, since before this stage is reached, nuclei which initially were distinguishable from one another have grown together into a single mass of silver. It is, however, very unlikely that even if every nucleus could be observed the general form of the curve in fig. 8 would be changed. 025: ee ERE to) 25 a5 a The most natural assumption to make is that longer develop- ment would merely result in an increase of the number of visible nuclei in proportion to the number already observable, and that this is the case is shown by the following experi- ment :—Two plates were given the same exposure under the wedge and partially developed, one for 15, and the other for 18 seconds. The values of No were then found for widely different intensities, with the following results :— (1) T= 1-00, No for 15 seconds development = 0°613, No for 18 seconds=0-980, whence (No) 18/(N,)15= 1°59. (2) I = 0°27, (No)15 = 0°153, (N,)18 = 0°240, whence (No) 18/{No)15=1°57 ; so that this ratio is practically constant, and the general form of the curve is indepen- dent of the development. |:00 a a 364 Mr. F. C. Toy on the Theory of the Variation of Grain Size. Heperimental. It will be convenient to deal first with the experimental curves. When the values of 2 were being found for the curve in fig. 4, the corresponding values for three larger sizes of grain were determined at the same time and in the same way. The characteristic curves for the four sizes are shown in fig. 9. | 100 80 60 3 —> 40 CROSS SECTION. a= 0-48 2. b = 1°75 ar. C =2:73hK. d = 3-934. 20 —|-0 —0-5 0-0 LOG Eom The important points in regard to these curves are that for one and the same emulsion :—(1) a set of large grains is more sensitive than a set of small ones, which confirms Svedberg and Anderson’s result (Phot. Jour. 1921, ixi. p- 325) ; (2) the characteristic curve for small grains has a greater maximum slope than that for large ones, 2. e., the ratio of the intensity which just changes all the grains to ' that which just causes the smallest possible change is larger the larger the grain size. As will be seen from the figure, the logarithm of this ratio for the smallest size grain is about 1:5, which is equal to an intensity ratio of 30:1, whilst for the largest size a ratio of 100: 1 is necessary to give half the curve. In Table IV. are given values of a, as read off the experi- Characteristic Curve of a Photographic Emulsion. 365 mental curves in fig. 9, corresponding to known relative intensities, and in the third column the values of A are TaBLe IV. Cross-section of (a) =0°98p’, : (0) = 175 pe . (c) =a (ome. “4 (2) == S305". | a (curve values). A. Relative | Intensity. (a) = =(6. =). @) (a). (0). (e). © @) 1-000 Oh 95'S; 9iae 980 24 o LGoO) ook 0-689 900 945 96:8 97:8 ou, 294, | 3°80 0-490 S607, 93077960 (97'0 204 266° 3:22 851 0-270 TO 6866 925. 4.04°5 Bo) 2°08 2:60) 2790 0-128 46:0. 7a:0.- SiO ~ S85 O62 loi 667 2-16 0-061 P24 525 O66 780 0D.) OFZ AAO FS 0-033 0:0. 26:0. 540. -676 G00 OFa0e.0°78 *) dei 0-016 C0. ,207., 405. 57-0 0:00 O22 052 0°84 0-008 0-0 TO. . 28:5. 465 0:00 O07 O83 90°62 0 0:25 0:50 0-75 1-0 I calculated. The A, J curves for the four sizes of grains are given in fig. 10. 366 Mr. F. C. Toy on the Theory of the Theoretical. Consider what is the effect of a variation in grain size on the nuclei distribution curve shown in fig. 1. We will first assume that the sensitivity of a nucleus is quite independent of the size of the grain in which it chances to be, i.e. once a nucleus is formed in a grain, its sensitivity does not change as the grain grows. This is apparently Svedberg’s assumption, for he says : ‘* the small and the larger grains in one and the same emulsion are built up of the same kind of light-sensitive material—just as if they were frag- ments of different size from one homogeneous silver bromide crystal.” If this is the case, then the only result of in- creasing the size of grain is to increase the total number of nuclei, and these will be distributed amongst the different sensitivities in the same proportion as before. Thisis shown in fig. 11, where the distribution curves for two sizes of Fig. 11. iy ie grain are given. We have made no assumption regarding the relation between total number of nuclei and grain size exeept that large grains have more than small ones. The curves relating I and No (average number of nuclei per grain) which will be obtained from distribution curves such as those in fig. 11 are shown in fig. 12. We have already shown that the No, I curve is identical in form with the A, [ curve, so that those in figs. 12 and 10 should be of the same form. As a matter of fact, there is a striking difference. The experimental curves in fig. 10 le practically parallel to one another at the higher intensities, and the point of in- flexion (which corresponds to the maximum ordinate in the nuclei distribution curves in fig. 11) moves towards the origin as the grain size increases. In curves (0), (c), and (d), fig. 10, which are for exceedingly sensitive grains, the point Characteristic Curve of a Photographic Emulsion. 367 of inflexion has moved so near the origin that the part of the curve to the left of this point does not show on the scale to which the curves are plotted. On the other hand, the theoretical curves in fig. 12 are characterized by the fact Fig. 12. Ls ie me that the ratio of the ordinates for different sizes of grain is independent of the intensity, and the value of I at the points of inflexion, I; and the average sensitivity do not change as the grain size is varied. Thus we cannot explain the effect of a variation of grain size on Svedberg’s assumption. Now let us assume that the sensitivity of a nucleus depends on the size of the grain in which it is contained, and that if Fig. 13. I oer 4 La it is in a large grain it is more sensitive than it would have been ina small one. The effect of this on the distribution curve for the larger grain in fig. 11 is to shift it bodily nearer the zero, thus decreasing the value of I; and increasing the average sensitivity, as in fig. 13. The Nol curves | 368 Mr. I’. C. Toy on the Theory of the plotted from these distribution curves are shown in fig. 14, and it will be seen that they are similar to the experimental curves in fig. 10. The reason why, for very sensitive grains, the lower half of the S-shaped curve appears to vanish (0), (ce), and (d), fig. 10, is that the value of I; is very nearly zero, but it would be shown if the points were plotted on a bigger scale. : Fig. 14, |= The evidence thus points to there being two reasons why large grains are more sensitive than small ones. Firstly, there are more nuclei present in the larger grains, so that a single grain has a greater chance of having at least one; and secondly, the average sensitivity of the nuclei increases with the size of grain. Svedberg in his most recent paper (2id.) discusses the relation between the average number of nuclei per grain and the grain size. He says:—“ The rapidity of the increase of the average number of nuclei per grain No with size of grain would depend on two factors: the ability of the developer to penetrate into the grain, and the homogeneity of the field of light in the grain. If the developer is not able to get into the interior of the grain, but only attacks the surface layer, then No would mean the number of centres in that surface layer, and therefore would increase in approximate proportion to the grain surface even in cases where the field of light in the grain was not homogeneous (because of strong ligbt absorption). On the other hand, if the developer is to penetrate the grain, No would depend upon the field of light in the grain. If the absorption of light were feeble, No would increase in proportion to the volume of the grain; if the absorption were very strong, No would increase approximately proportionally to the cross-section of the grain.” Later in Characteristic Curve of a Photographic Emulsion. 369 the paper he compares the variation of Ny with grain size for grains which have been exposed to light with the variation when the exposure is to X-rays, and suggests certain deduc- tions as regards the absorption of light ‘and X-rays by the silver halide from the difference which he finds. Now, from fig. 10 we see (since A= Ny) that the manner in which No varies with grain size depends on the intensity to which the grains have been exposed ; we can select an intensity such that No varies in almost any manner we please. Thus, unless the difference between Svedberg’s results and those found here is due to the different emulsion used, there seems to be no justification for making deductions from the relation which is found between No and the size of grain at one fixed arbitrary exposure. The theory which has been advanced here is capable of explaining an important fact which appears quite inexplicable on such a theory as Allen’s (2bzd.). Itis well known that the sensitivities of the grains in an emulsion depend to a great extent on the conditions of precipitation and ripening; and that, in different emulsions, sets of equal-sized grains may have quite different sensitivities, and even different maximum slopes for their characteristic curves. If, as Allen suggests, the nucleus is really a simple molecule of silver halide which has lost an electron, its characteristics will be the same whatever the emulsion, and it is difficult to see why grains in one emulsion should be more sensitive than those of the same size in any other emulsion. If, however, the nucleus is not silver halide, it is very probable that the conditions of pre- cipitation and ripening do play an important part in deter- mining its characteristics. Thus, on Renwick’s theory, the condition of the colloidal silver which is produced will certainly depend on such factors as the kind of gelatin, conditions and time of ripening, etc., and the ease with which colloidal silver particles can be coagulated will be affected by the amount of gelatin present, since this is a protective colloid. The creat difficulty in accepting Renwick’s theory as it stands is this :—It is known that an unprotected silver sol is very stable to the action of light. Therefore, if a protective colloid is present, it will be still more difficult to effect its coagulation and precipitation by light, whereas in the case of our most sensitive silver halide grains the energy necessary to make them developable is exceedingly small. Liippo- -Cramer (ibid.) believes that the mechanism of the formation of the latent image is not the same for the most sensitive and very insensitive emulsions, and he claims that Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. 2B rs 2 one. 370 =Characteristic Curve of a Photographic Emulsion. this is supported by his experiments. He found that the sensitivity of a very fast emulsion was decreased considerably by treatment with chromic acid, but that the sensitivity of a very slow emulsion remained unchanged. He explained _ this by the existence on the surface of the sensitive grains of colloidal silver, formed during the ripening process, which was not present in the insensitive grains, and which was removed by the chromic acid. It is very difficult to imagine that the Fudacreneal hight action varies with the kind of emulsion, and that considering a whole series of emulsions, from the most sensitive to the most insensitive, there is a transition region where an entire change of mechanism takes place. Strong evidence against Liippo-Cramer’s view is that Svedberg (zbid.) has shown that in one of the slowest emulsions the reduction centres are distributed amongst the different grains according to the same law as has been shown here to hold for their distribution in the case of one of the tastest commercial emulsions. This is in favour of the view that for all kinds of emulsions the process of the formation of the latent image is the same. The existence of this chance distribution of developable “centres” in the grains does not conclusively prove that they are the kind we have considered in this paper, and there are at least three other possibilities. Assuming a discrete structure of the radiation, the centres may, as suggested by some, be the points of impact of light quanta on the grains, but the fact that the majority of these centres are located on the edges of the grain is strongly against this view. Also within the crystal there may be a chance concentration of the hight energy at certain points, and both these possibilities are being tested in this laboratory. Again, this chance dis- tribution may be due merely to the fact that the grain as a whole is changed by the light, but the developer reaches some points of it sooner than others. If this is so, there appears to be no reason why the average number of centres per grain, considering only developable grains, should in- Increase, as it does, in a regular manner with the light intensity. The author believes that the evidence so far obtained is mainly in support of the theory discussed in this paper. In conclusion, the author wishes to express his thanks to Dr. T. Slater Price, Director of Research of the British Photographic Research Association for much _ valuable criticism and advice. _ On the Stark Eqect for Strong Electric Fields. 371 Summary. A theory is advanced which explains the relation found experimentally between the number of geometrically identical silver halide grains.made developable and the lght in- tensity. It is assumed that there exist in the grains particles which are not silver halide, and which are formed during precipitation and subsequent ripening. With any normal exposure (?.e. one which gives a value between 0 and 100 for the percentage of developable grains), it is these particles which form the reduction nuclei, the only action of the light being to change their condition in such a way that they become susceptible to the action of the developer. Each nucleus does not necessarily require the same intensity to changeit. The nuclei are scattered haphazard amongst the grains according to the laws of chance, and only grains which have at least one will be developable. The sensitivity of a grain is the sensitivity of its most sensitive nucleus. The effect of a variation of grain size is explained, and it is shown that Svedberg’s assumption regarding the similarity of the light-sensitive material in large and small grains is not in agreement with the experimental facts in the case of a fast emulsion. XXXII. On the Stark Hyfect for Strong Electric Fields, To the Editors of the Philosophical Magazine. GENTLEMEN,-~— M* attention has been drawn to the results of experi- ments by Takamine and Kokubu™* in which an effect of the nature indicated in a recently published paper t of mine was detected, namely, a shift of the central line in the perpendicular component of Hy in a strong electric field. Before comparing the experimental amount of this shift with the theoretical value it would have on the Quantum theory of spectral lines, it is necessary, however, to point out a slip in my paper referred to above: thus on p. 945 a term is missing from the value of the contour integral (4), instead of (6) the full value should be a os Diatval? SBD? /, ras * “The Effect of an Electric Field on the Spectrum Lines of Hydrogen,’ Part III. Memoirs of the College of Science, Kyoto Imperial University, vol. ili, p. 271 (1919). + Phil. Mag, xliii. May 1922, P. ae this will be referred to freely. a - — 372 On the Stark Liffect for Strong Electric Fields. Consequently the third term on the right-hand side of equation (25) p. 948 should be ain WY? N’ N’i (ji 1024m°H'n,ie * (n)—N'(m)}, » 0. 2 Cap where N’ is now given by N’(n) = (ny +7243)! 35 (m+1413)° N(n) MW |e oh Se Pe Soi 2 i ee 15n3° — 21 (n,g—7) OF ee N(n} being still given by equation {10). In view of the. identity N(n) = (21 +13) (629? + 6rgn3 + 13”) + (2ng + nz) (62)? + 6nyn3+ N57) = 3(ny +n +73)? —3(my + Ng +73)(Ng— 0, )? —n3?(ny +ng+nz), N’ can be reduced to the form N'(n) = (1 +724+7n3)4 £17 (ny + not ng)? — Ing? —3(»—n1)?}, Gin). which shows in conjunction with (i.) that the remarks in the paper about the symmetry of the components are not affected by this correction. In order to calculate the amount of shift of the middle n-component of H, we observe that this component can arise from any of three possible transi- tions corresponding to (ms, Mg, My 3 3, Ng, M4) =i (3, 1,15 2;05-0) sor s252 52,10, 0) ori 2 2a respectively, the values of {N’(n)—N'(m)} corresponding to these combinations being — 2°14 x 10°, —2°59 x 10°, and again —2°59x 10° respectively. And on substituting the values of the universal constants in (1.) for hydrogen (H=e) the following expression is obtained for the wave-length shift } 2 2,12 An =—™ Ay =—1-42™" {N’(n) —N'(m)}. C : Damping Coefficients of Electric Circuits. ato This gives for H, (A =4°34x 107°) and the value of F used by Takamine and Kokubu *, namely, P4833 * 10° ¢.¢.s.e.s..units |-=1°3.x 10° volt.x ¢m.~"], An = °36A or *43.A respectively. The experimental value observed by Takamine and Kokubu is about 1A, which is larger than that predicted by the Quantum theory. It is, however, possible that part of the experimental shift is due to a Doppler effect, and in any case the experiments could hardly be considered accurate enough to exclude a possible experimental error of what is only about 4 an Angstrém unit. On the other hand, the photographs of the shift [plate 1. fig. 1] pownt decidedly to a general displacement of all the components in the direction pre- dicted by the theory, namely towards the red, and may be taken as corroborative of at least this qualitative aspect of it. It is seen from (i.) and (ii.) that this lack of symmetry in a strong field would be expected on theoretical grounds to be more pronounced for the higher members of the Balmer Series (e.g. H; or H.), and it would be highly desirable ta obtain measurements relating to these lines as a further test for the quantitative aspect of the theory. In conclusion I wish to thank Mr. W. E. Curtis for drawing my attention to the experimental results already referred to. King’s College, London, Yours faithfully, May 12th, 1922. A. M. MosHarRaFa. XXXII. On the Damping Coefficients of the Oscillations in Three-Coupled Electric Circuits. By i. TAXaGIsHt, Electro-technical Laboratory, Department of Communica- tions, Tokyo, Japan tf. HOUGH the importance of the problem of three-coupled electric circuits has arisen with reference to radio- telegraphy, it does not seem to have been attacked with any great amount of attention except by B. Mackt, EH. Bellini, _and very recently L. C. Jackson{. The valuable paper of the latter made me feel very much interested, especially as it will make an important contribution to radio fields, but ¥ 1. &. + Communicated by the Author. t Phil. Mag. vol. xlii. No. 247, July 1921, p. 35. o74 Mr. E. Takagishi on Damping Coefficients of unfortunately I found there slight errors concerning the damping coefficients of the circuits. Now, let us proceed to correct them, using the same notation and abbreviations for the sake of simplicity. Comparing coefficients in the. equations (5) and (6) in his original paper, we obtain, instead of (7), —2(q+r+s) = (Riles +L, ReLs+ LL.B, — R,M.,;? Ih, oM 33? ar Jal es? = D, : G. ) | (wo? + @;?+ @3”) 7 as J Lyle = Mie" at LL; — M3? a LL, = Moo | i 13 | C, "2 +TTQR,+ RRs +R Rls b=D, . Gi) —2{o(r+s) +0°(s+9) +03°(q+7)} y { alt Le a AE Ey ane LF, “ll 4 | | seesoatelg he Go — (@1°@9" + @9°@;," + @3°@ 1") dey eee we Ee f ce Co OC, Eerie iS ee) tau A a2 ; alin Gy a GC, ) —D, . (iv.) | \ Ls — 2( 0)?" 8+ @,"03"9 + @;°01"7 ) Ry lay ai \ | = alee + — GC, 4- Ee, —)D, 5 ‘ rah 5 6 (v.) (7') ~@, 773 | IL Se > 3 (Wile) where D = My» Mos Ms; ara L,L, Ls + L,M,.?+ LoMs,?+ LsM,.? . (vii.) J From equations (7") li., 1v., v1., vil. we obtain the same equation for w, as (8) and (9) i in his original paper. For the damping coefficients g, r, and s, however, we find the following values, different from those in (LO je | Oscillations in Three-Coupled Electric Circuits. 375 Now, making use of the abbreviations given below and in the original paper, we get the following equation : A —2g+r+) =X, 2 2 : B — 2 { g(a? + 3”) + 7(@3? + @,") + 8(@)" + @,.”)} = ae C —2{o02o/7tofo"rt+oa,"s} = Xx? where A =k (1-8) +4o(l—y?) $hs(1—2), B= n?(ky thy) +P (hy + hy) + m? (ky + hes) + ky hohs, C= khym?n? + kl?n? + kyl? m?, X = — (1-2? — 8? —9? —2aBy). Solving these simultaneous equations, we have Q = YY? ea: =y aS Q Tey? in which A ee) Lee) x 9 2 A 1 it 4 2 —2(w; + @, ay — (w +o.) nm B (37+ ,”) (a? + 2”) — 2(@3"@,’) = 2(@ >a") CG W3°@ 1" W?W9" z “ fo) i hae Bae’ CY, similarly ge © (oP -o2) haa Seem PEC}, se = (o2=o,) [4a — Bay + 0}, and Y = 8(@,2?—@,”) (w2?— 37) (@3? — @)2) 376 Prof. 8. C. Kar on the Electrodynamic That is, q = [ thym?n? + kan? P + hl?) — @? {PF (he + kg) +m? (kg + ky) + ?(ky + hy) + keykoks} + or" {ky (1 — 8?) + ko(1—y’) + ke3(1—a?) } | | =- 2 (1—2«?— B89? — 2a Bry) (w1? — 0,7) (@;7—@3"), == | (kym?n? + kon?l? + ksl?m?) — wo? {0 (ke + ks) -+ m? (kz + Ay) +n? (ky + ke) + kykoks} + wo*{ky(1—B?) + ky (1—y¥’) sii) | 21 = 27 — 6° — 9? — 2a By) (os 03 | es oe = | (kym?n? + kon?l? + kgl?m?) — ws? LP? (ky + kg) + m?(hg + fy) +n? (hy + ke) + kyhkoks} + @3* {ky (1—?) + ko(L—y?) hela") ta 22a payee By) eae) oe eee ~— (7) On inspecting these equations for damping coefficients it is noticed, at once, they are also correct with respect to the dimensions. XXXIV. On the Electrodynamic Potentials of Moving Charges. By 8. C. Kar, M.A., Professor of Mathe- matics, Bangabast College, Calcutta *. VENUE electrodynamic potentials of a moving charge or the electron have been the subject of several in- vestigations and the earliest were those of Liénardt and Wiechert f. Among recent writers who have found the potentials on a relativity basis may be named Sommerfeld § and M. N. Sahal|. Both of these writers performed a four- dimensional integration in the Minkowski space-time mani- fold and have obtained results which are quite general. It appears to the present writer that the Liénard and Wie- chert result—and the method admits of easy extension to the case of a straight linear current—may be obtained easily enough by a Lorentz transformation to a rest-system and back without resort being had to four-dimensional integration. * Communicated by the Author. + L’ Eclawrage électrique, vol. xvi. pp. 5, 53, 106 <1898}, { Arch. Néerl. vol. v. p. 549 (1900). § Ann. d. Phys. vols. li. and liu. || Phil. Mag. vol. xxxvii. p. 847 (1919). Potentials of Moving Charges. 377 The equations for the potentials may be written BE). (iceb, F, G, BH) =pox(ic,u, .v, w) where eal, yes ae +10? 0° cM ees. role and UO = Pane an + Oy Tee It is well io point out that this mode of writing the equations is Henne different from the customary mode where OF’= =pok so that our F is cK’. This deviation from usage 18 amited by the greater symmetry and homo- geneity of form resulting. The equations for (magnetic intensity) and d (electric intensity) will on account of this change assume the forms Hee ene EL), c So ae d=— alh, G, i) + (55 Oy? dz (2) It is evident that the operator 1 is an invariant under a Lorentz transformation. It will therefore follow that - (ic®, F, G, H) is a four-vector, because x(ic, wu, v, w) is a four-vector. Therefore DSi — Fx — Gdy—H82 which represents the scalar product of the four-vectors (ic®, Ff, G,H) and (icdt, da, dy, dz) is invariant under a Lorentz trans- formation. Therefore, e’@dt — Fbx— Gov — H$z=c?@' bt! — F'82’ — G'dy' — H'62", where the dashes refer to a system of axes moving with velocity v along the axis of «. But bu=x(du' + vot’), déy=6y’', eck and bts a( de + = pt z =} where c=1fy/ 1-2. = Substituting and equating coefficients of 842’, dy', dz’, and ot' we have F'=<(F —»®), G'=G, H’=H, and o'=(@-"2), 378 Prof. 8. C. Kar on the Electrodynamic These formulee are exactly similar to the usual formule: for 6x, dy, 6z, dt and connect the potentials for any system of axes with those of another moving with velocity v along — the axis of 2. The reversing formule are H=«nci’ fo’) Ga G7, A= HH’, and O— (Oe vl! =} (3) Let us suppose an electron moving with velocity wv along the axis of w and let us take a system of axes moving with the electron. It is apparent that for the latter system of axes the electron is i. rest. The vector potentials. CE", GE’) =0and O'= i r! is the distance of the point P at which the potentials are considered. For the original system of axes, therefore, we should have according to the formulee of transformation given above, B= xy - —, G= 0, ia) and = eer. An Agr r’ is, however, expressed in terms of the coordinates of the rest system and it will be necessary to transform it to a form involving the coordinates of the original system. But if the time-difference between the point P and the electron is Ai’, then be = cAt’ =f rE ar| 1 ee C Cin arial due toa static charge e where € and Da a which are Liénard’s results. ‘é Potentials of Moving Charges. oto (4) Let us suppose a straight linear uniform current to arise from continuous and uniform rush of electrons in the conducting wire in the direction of the current. Viewed from a system of axes moving with the common velocity of the electrons the phenomena reduce, as far as the rushing electrons are concerned, to the case of a linear and uniform distribution of electric charge. If N electrons each with charge —e be supposed to rush with velocity v to the observer in the rest-system the linear density of static charge is —Ne. From the ordinary theory of potential, the potential ®' for such a distribution is —2Nelogr’ where r?=y"+2” and (F", G', H’)=0. Transforming to a moving system according to our formule we should have Poe ——26Nevu loo'*) “G—G'=—0,; B= 0, G= 70! = —2eNelogr [n= |. The magnetic field therefore would be given by h,=0,. InNevz 2nNev hy=— oi ;h=+ aE / and the electric field would be Cr 79 InNez amen by d_— 0, d,—— ees dj=— at weet the veon- ducting wire, however, there is also a linear distribution of positive nuclei at rest of which the potential would be +2Nelog r. The electric field due to these would be given by d,’=0, IN Vine d= + ~ eeu d= Pith 7a r The resultant electric field would therefore have the 2Ney 2Nez 2 a Ch a) D) ( The magnetic field is of finite magnitude and cir- 1—«), andisof the order of —. ec? 2nNev ae cular round the wire, the resultant being ———— which is : cr ae : ee (eurrent):.. quite in accord with the expression eneonreny) if we put fe kNev Nev . cM current =——— or —— neglecting quantities of the order c c 9 “ .. : ; , In comparison with unity. c ee EE SS Ee F 380: XXXV. The Identical Relations in Hinstein’s Theory. By A. KE. Harwarp*. VQYHE March number of the Philosophical. Magazine con- tains an interesting proof of the identity Gi eee pve Otp by Dr. G. B. Jeffery. Apparently it is not generally known that this identity is a special case of a more general theorem which can be very easily proved. I discovered the general theorem for myself, but I can hardly believe that it has not been discovered before. The theorem is (Buvo®)r + (Buor®)y + (Burr?)o= 0. . . C1) This identity can be verified in a rather laborious manner by forming the covariant derivative of Byys?, but it can be more easily proved as follows :— The identity Au, Vo = Jai ov — Buvo® Ap e e ° ° (2) can be easily generalized so as to apply to the case where instead of the vector Ay we have a tensor of any order ; thus Aw, OTe Any, r= Buyer? A pv + Byer? Anup: This is proved in the same way as (2). Now, if Ay be any covariant vector, then (Ap, vor — Ap, vro) + (Ag, ory — Ap, ovr) + (Ap, rxo— Ap, rev) = (Ay, ve—Ap, ov)r + (Ap, or —Ap, ro) v + (Ap, ro — Ag, v7) 53 ~. Byor? Ap,» t+ Byer? Ap, p+ Bury? Ap,o+ Bory? Ap, p + Buvo® Ap, r+ Brro® Ap, p = (Buyo? Ap)rt(Buor? Ap)y + (Bury? Ap)o- * Communicated by the Author. The Identical Relations in Hinstein’s Theory. 38h Now, (Byvo® Ap)r = (Bure?) Ap + Bure? Ap, z ; so after cancellation we get (Byor? + Bory’ + Broo”) App = [(Buvo)r + (Busr?)» + (Burv®)o} Ap. The expression in brackets on the left vanishes identically. Since Ap is arbitrary, the expression in brackets on the right must also vanish. Q. The identity Byer? a Dery’ + Brve® =.) follows at once from the well-known identical relations: between the Riemann symbols. The three-term identity is. usually stated in the form (utov)+(yorvr) +(pvtc) = 0, or in the modern notation Buyvor+ Burvo + Buory =i) 3 here Buror denotes gre Buyos = (tov). Since Buryvo=Byoyr and Buory = Bopyr, 0= Buyer ap Byour rE Boor = Gre (Buro® “la Bropf aT Bopr*). We assume that the determinant | gy,| = g does not vanish in the region under consideration; therefore the expression in brackets must vanish. - This identity can also be proved by observing that the expression ee ee NRL e tee e vanishes if A, is the derivative of a scalar; for in that case Ay.or = Ac,vr, Ao,ry = Ar,ov, and Az, ve = Av, ro. If we contract (1) by putting T=p, we get (Biot at Ops,v—Ge,c = 0... 2°. (8) 082 Mr. H. 8S. Rowell on Hnergy Phrtition If we contract this again by multiplying by ne we get the familiar identity 2G. ee ogi iy for since (g'”)o = 0, 9” (Bune? )p = (gt? Bura?)p = (gH gP" Buvor)p = (9? og!” Brovu)p = (7°" Brow”) p == (O° Gras — Gee ; ‘similarly gt? (Crug, v) =P? Gael = Ge, and cas (Guy, 9) Py (gh Guy) o = (G)o == ‘since G is a scalar. Jersey, 18th May, 1922. XXXVI. Energy Partition in the Double Pendulum. By H. 8. Rowe. * N a letter to ‘ Nature’ (July 28,1921) the present writer gave a theorem on the double pendulum which is capable -of interesting extension. If the masses of the bobs are m and M and the respective amplitudes are a and A with suffixes.to denote the normal modes, then the theorem states that AOR ans If this equation is squared and Boae sides multiplied by am?/M?, we have M172, MAg?N9" MA,?n 1 MA,?n,? = where 7; and ny, are the radian frequencies of the two moles. ‘This equation may be readily interpreted thus :— “The ratio of the kinetic energy of one bob to that of the * Communicated by the Author, in the Double Pendulum. 383 other bob in one mode is the reciprocal of the corresponding ratio in the other mode,”’ Proceeding to the general case of an elastic system with , . 5 Ls . two degrees of freedom, using Professor Lamb’s notation, 2T = Aé?+2HOd + BE», 2V = 2V)+a0? + 2h0gp+ bd’ ; so that with a time factor = in (N04 Hh) = ad +hd, n*(HO+ Bd) = h0+b¢ 5 ' bH—AB whence the product of the roots in @/¢ is — RAS If H=0 so that T is a function of squares of velocities, the product of the amplitude ratios is —B/A, or, in the double pendulum, —M/m. If h=O0 so that the potential energy is a function of squares of displacements, the product of the amplitude ratios is b/a, i.¢. the ratio of the two stability coefficients. ‘Thus in either case we have an energy relation. For the kinetic energy take ho and). —— = — = square and multiply by A’/B?, and insert the frequencies. For the potential energy take A=0 and —1?=—- which, when squared as before, yields a similar relation. The two results may be expressed in words thus :— When the Kinetic or Potential Energy is written as a function of squares only, the ratio of the Kinetic or Potential Energy expressed in one co-ordinate to that expressed in the other co-ordinate for one normal mode is the reciprocal of the corresponding ratio for the other normal mode. This investigation gives an insight in certain cases into the indeterminateness of the normal modes with equal periods. [ 8eiee XXXVII. Velocity of Electrons in Gases. To the Lditors of the Philosophical Magazine. GENTLEMEN,— |e a paper in the Jahrbuch der Radioactivitidt und Electronik (vol. xvill. p. 201, April 1922) H. F. Mayer gives an account of some of the formule obtained by different physicists for the velocity of ions or electrons in gases due to un ‘electric force, and concludes that a formula recently given by Lenard is more correct than the others. Among the other formule which are discussed, the author gives what purports to be an account of a formula for the velocity of an ion which I published in the ‘ Proceedings of the Royal Society’ (A. vol. Ixxxvi. p. 197, 1912), and states that this formula is so incorrect that it does not even give the right order of the velocity. JI should like to draw attention to the way in which Mayer has misinterpreted the matter, and to quote the formule as I gave them for the different cases in which the mass of the ion is small or large compared with the mass of a molecule of the gas through which it moves. On pages 199, 204, and 206 of my paper, three formulee are given for the velocity U of an ion in the direction of the electric force X in terms of the mean free path / of the ion, its mass m, charge e, and velocity of agitation uw which is supposed to be uniform and large compared with U. The first of these is UHXlimu,, . . . and applies only to cases in which the mass of the ion is small compared with that-of a molecule of the gas (an electron for example), since it is here assumed that after a collision with a molecule all directions of motion of the ion are equally probable. I pointed out that when the mass of the ion is larger than that of a molecule of the gas, all directions of motion of the ion after a collision are not equally probable, and that in this case an ion travels a considerable distance (having an average value 2X) after a collision in the direction in which it was moving before a collision. A more general formula for the - velocity was given, which is Us XeG@Pymu. 258 4 een If the mass m of the ion is so large compared with the Velocity of Llectrons in Gases. 385 mass m’ of a molecule of the gas that all directions of motion of a molecule become equally probable after a collision with an ion, it was shown that formula (2) reduces to MAU Wey a We Mien! area ose, (3) as in this case it may be seen that t+ he Three . & . e ° . (4) It will be observed that formula (2) reduces to (1) when Xr is zero, that is when m is small compared with m’, so that either of these two formuls: may be applied to the case of an electron moving ina gas. Mayer, however, selects formula (3) to find the velocity of ions of small mass or electrons, although it is definitely stated in my paper that formula (3) refers to large ions, and the relation (4) on which it depends ean only hold when m is greater than m’. As the correct formula (1) for electrons differs by the factor m’/m from formula (3), it is unreasonable to expect the latter formula to give the velocity. of an electron. The above formule, obtained by simple considerations when the velocities of agitation of all the ions are taken as being the same, are of course not absolutely exact. There is a numerical factor by which the expressions should be multiplied in order to allow for the variations of the velocity of agitation about the mean velocity. In the most interesting cease, which is that of electrons moving in a uniform electric field, the value of the numerical factor is about °9, but it has not been determined exactly. The determination of this factor is very difficult, as the distribution of the velocities of agitation of the electrons depends on the energy of an electron which is lost in. a collision, and experiments show that the proportion of the total energy of an electron which is thus lost depends on the velocity. This problem has been fully considered by F. B. Pidduck (Proceedings of the London Mathematical Society, ser. 2, vol. xv. pt. 2, 1915), who shows that under certain conditions the proportion of the velocities which differ largely from the mean velocity of agitation is much less than the proportion indicated by Maxwell’s formula for the distribution. It appears that the error introduced by taking the velocities of agitation as being all equal to the mean velocity may be no greater than when the velocity distribution is taken as being the same as that given by Maxwell’s formula. In order to obtain an exact formula for the velocity U it Phil. Mag. 8. 6. Vol. 44. No. 260. Aug. 1922. 2C 386 Prof. H. A. McTaggart on the Electrification would be necessary to take into consideration the variation of the mean free path of an electron with its velocity of agitation, and the large reduction of the energy of an electron when ionization by collision takes place. These points in connexion with the motion of electrons in gases have not been taken into consideration by Lenard, and it does not appear that his formula is more correct than others which have been proposed. Yours faithfully, 8rd May, 1922. ) JOHN S. TOWNSEND. XXXVI. On the Electrification at the Boundary between a faquidand a Gas. By Professor H. A. McTaccart, M.A., University of Toronto”. | ANY years ago, in the course of some experiments on 4 the effect of an electric current on the motion of small particles in a liquid, Quincke (Ann. d. Phys. exiii. p. 513, 1861) observed that small gas-bubbles in water moved as though negatively charged. Although a good deal of atten- tion has been paid to the movement of solid and of liquid particles in such cases, very little effort has been devoted to the study of small spheres of gas suspended in a liquid—one obvious reason being the difficulty of controlling them while under observation. A systematic examination of their elec- trical properties ought, however, to yield further information as to the physics—and chemistry too—of surface layers. | Before the war experiments in this field were begun by the author in the Cavendish Laboratory under Sir J. J. Thomson, and some results were obtained. Measurements were made (Phil. Mag. Feb. 1914, p. 297) of the velocity, uuder a fall of potential, of small spheres of air in distilled water and their electrical charges were estimated. ‘The effects on the charge of the addition of minute amounts of various inorganic electrolytes were studied. Results were obtained (Phil. Mag. Sept. 1914, p. 367) showing how the charge varies with the presence in the water of certain alcohols and organic acids, and a parallel was shown to exist between the variation of the electric charge and the surface tension. The present paper deals with some further experiments carried out in the University of Toronto, and describes the variation observed in the electric charge on small spheres * Communicated by Professor J. C. McLennan, F.R:S. : at the Boundary between a Liquid anda Gas. 387 of air when a particular electrolyte, Thorium Nitrate [Th(NO,),], was dissolved in water. This salt was selected for special study because it had been found to be unusually active in charging these surface layers. The apparatus used was similar to that referred to in a former paper, one or two changes being. made in i ton greater convenience. The arrangement is shown in fig. 1. A isa small cylindrical glass cell rotating about its axis on pivots and driven by a belt of thread from a pulley F on a Rayleigh motor. This motor was made in the laboratory workshop, und has, instead of the usual fly-wheel with a hollow rim filled with water, a solid brass wheel H—a modi- fication suggested by Professor Wilberforce of Liverpool. The wheel, although loose on the shaft, has enough friction, when a heavy oil is used for lubricant, to keep the shaft in steady motion after synchronism with the tuning-fork is attained. D is a timing device consisting of a vertical post carrying a pointer and made to rotate by a toothed wheel working in 202 388 Prof. H. A. McTaggart on the Electrification the worm HE. The pointer rests by its own weight on the top of the post, but at any instant in its motion over the fixed dial D it may be raised and stopped by a small electromagnet controlled by the key B. When released it falls back on the _ post and begins to record time with the same regularity as the tuning-fork. It forms a very convenient stop- -watch if velocities are to be measured. A travelling microscope M measures the distance travelled by any bubble on the axis of the rotating cell. The water used was twice distilled—the second time in ‘“¢ Pyrex” glass and condensed in a silver coil. The thorium nitrate was by Merck, and-was assumed to have 12 H,O—water of crystallization. A stock solution was made up containing 4 x 10° equiva- lents per c.c. (1/250 normal), and from this other solutions were made by successive dilution. A first series of readings was taken with various concen- trations of the salt, but with bubbles of nearly the same size in order to reproduce the effects previously observed—the method of working being to fill the cell A with the desired solution, introduce a single bubble of air with the gas pipette, and set the cell in rotation. The bubble very soon takes up a steady position on the axis, and its motion under any fall of potential L. may be examined. Very small concentrations sufficed to reduce to zero the natural negative charge found in pure water and to give the small sphere of air a positive charge. The following readings are typical :— Fall of potential ......... 34 volts per cm. Diameter of bubble...... 0-3 mm. Concentration. 4: Velocity of Equivalents sea bubble. per C.c. aoe ems./sec./volt./em. ASC TO Ot Mak se teats + 5x10-4 SLOT Oy ohare phate a - slower 8510-9 | Lae + very slow (CEES MO) DT VOT?) 5s eens ~ slow ACTEM O! 1: A ee | — faster ASN Ogre 5. le — faster Pure’ waver .........0e% _— 4x1074 ' at the Boundary between a Liquid and a Gas. 389 The zero point was reached at a concentration of about 7x 10~, a result rather higher than that given in a former paper. The salt was an entirely different sample, and may not have contained the same proportion of water of erystal- lization. (See Abegg and Auerbach, ‘ Inorganic Chemistry.” A series of readings was then taken for spheres of air of different sizes, one ‘object being to observe the charge on very small spheres. It is very difficult, by the use of any kind of pipette, to introduce into the rotating cell bubbles smaller than 1/5 mm. in diameter. To avoid this difficulty the following mode of working was adopted. ‘The solution was first placed in a partial vacuum to remove as much dissolved air as possible, and afterwards poured inte the cell. A bubble into this gas-free solution slowly decreased in size by absorption until it vanished, while the electric charge could be observed at any stage. Under these circumstances it was found that for a suitable concentration of solution a sphere of air which began with a small negative charge almost invariably and in a regular way reduced its charge to zero, and gradually took on a positive charge. ; The following readings illustrate this point — No: SRE oy cee ethers - = apne per c.c. In mm, =: WO Sea tees eee 0:26 — 0717 = 0°14 — 0°10 0 0:08 No. 2. i ae ae a oO cee per C.c. in mm. LO a st oe 0-44 - 0:35 me 0°26 — 0-17 0 0°14 390 Prof. H. A. McTaggart on the Electrification Novia: Concentration. Diameter of giasvae Equivalents sphere ase x per’e.e. in mm. 107? bo Le ee 0°62 — 0°53 — 0°39 ap Oat7, + 0:08 + No. 4. , Concentration. Diameter of Sign of Equivalents sphere charge. per c.c. in mm. LOR < OMe ese: 0-71 = 0°53 — 0°44 — 0°35 0) 0°32 a 0:23 + No. 5. yore ere ae per c.c. in mm. LO? OMe clues see 0:28 = 0-17 — 0:08 _ 0:05 — It will be seen from the first four examples given that at a concentration of 10-°x 5:7 the change of sign occurs in every case. Rarely, as in No. 5, and then only when the original sphere was small, did the sign remain the same. Even then the charge grew steadily less. In_ practically every case the negative charge slowly decreases as the bubble gets smaller, passes through zero, and increases to a small positive value. at the Boundary between a Liquid and a Gas. a9] Three examples are given for slightly greater concentra- tions :— No. 6. Concentration. Diameter of gi f Equivalents sphere raved : charge. per c.c. in mm. % TOF GG ek 0°53 = 0-41 0 0°35 at 0°17 + 0:08 +- Nada hh Concentration. Diameter of ; : Hquivalents sphere me o per c.c. in mm. Be. TO? 660.2! i 0°35 = 0:28 0 0:26 ae O17 + 0°14 + No. 8. Concentration. Diameter of d Equivalents sphere Bee a per ¢.c, in mm, ean We? iS ees te ita 0°44 Almost zero. 0°35 —- 0-17 + Above a concentration of 10~* x 8 the bubbles were always positive. The examples given show that the spheres do not all have the same size when they reach the zero—isoelectric—point in a given solution. The larger a sphere is at the beginning the larger itis when its charge becomes zero. This suggests, as the cause of the change in sign, a kind of coagulation of something in the free surface. It is known that, in a solution of thorium nitrate in water, 392 Prof. H. A. McTaggart on the Electrification hydrolysis occurs with the formation of thorium hydroxide thus— Th(NOs),+4 HOH—>Th(OH),+4 HNOs. There is present in the solution some of the original salt, some acid, and the hydroxide in colloidal form. The pre- sence of the last-mentioned was suspected as one of the causes producing the reversal of sign, and experiments were then made to test its activity in altering the charge. A colloidal solution of thorium hydroxide as free as possible from salt and acid was prepared by dialysis (Burton, ‘Physical Properties of Colloidal Solutions,’ 2nd Ed., p. 16). A dialysing “‘ sleeve’? shaped in the form of a test-tube was made of “ parlodion”’ (sold by the Du Pont Chemical Co., New York}. A solution of the parledion in ether and alcohol was used to coat the inside of a test-tube of suitable size. After the solvent had evaporated the parlodion re- mained as a thin but strong film which when detached from the glass served very well as a dialysing vessel. For this experiment a solution containing about 2 gm. of salt in 50 c.c. of water was dialysed for a period of three weeks, after which an estimate was made of the colloid pre- sent. A sample of 10 c.c. evaporated over sulphuric acid gave a residue of -0034 9m. The residue formed a thin layer of gelatinous material on the bottom of the evaporating dish, with drying cracks across it in all directions. The effect of this colloid on the charge on small spheres of air in water was then examined, the dialysed solution above mentioned being diluted as shown in the following examples :— C.c. colloid Diameter of No. solution in sphere pier of 100 ¢ c. water. in mm, rae Ate Siac Bas 10 0-21 te 0-12 + 0:07 ae DN ge Borhihioure 5 0-17 aK Beene ss ral 25 0-17 + 0:07 + It is seen that the surface is charged positively by the presence of very small amounts of the colloid. The following examples show the gradual reversal of the at the Boundary between a Liquid and a Gas. 393 sign of the charge accompanying the absorption of the bubble :— C.c. colloidal Diameter of : No. solution in sphere ele of 100 ¢,c. water. in mm, mare ise CT Al A a8 1:0 0-17 = 0-14 + DMiereetecs 0-5 0°35 — 0°26 -- 0:12 “ins 0:05 se Oi A erceahd 0:25 0°32 _ 0-17 — 0:08 — 0:05 se The experiments show that the colloidal thorium hydroxide gives both the effects observed with the ordinary solution. It not only charges the surface positively if present in sufhi- cient amount. but it also exhibits the reversal of charge with diminishing size of the bubble, and this, too, in concentrations of thorium of about the same order as in the case of the salt. TNiscussion. The ‘state of the matier and the nature of the electric forces in surface layers of liquids is still a subject on which no very clear ideas exist. Hxperiments on electro-endosmosis all point to a selective action in such layers so far as the ions in the solution are concerned. But the observations are always complicated by the presence in contact with the liquid surface of a solid whose role in the selecting we are ignorant of. The same is true of cataphoresis experiments with solids, as, for example, in the study of the electrical ‘charge on colloidal particles. This difficulty is avoided, however, in similar experiments with small spheres of air— or any gas—and in such cases we can safely regard any effects observed as due largely to the properties of the liquid and its free surface. In particular, the electrical charge existing at any air-liquid surface may be considered as the result of forces residing altogether in the liquid. It ought to be possible, then, in considering potential differences at solid-liquid junctions to isolate the contribution of the liquid. In the case of thorium nitrate in solution the selective 394 Electrification at Boundary between Liquid and Gas. action of the air-water surface is very marked, a positive charge being acquired by the surface with very minute concentrations of the salt. The positive ions available for selection are Tht and H™, but neither of these separately can be responsible for the unusual activity of the salt. ‘The mere presence of H™ ions, as, for example, in the form of an acid, does net produce so great an influence on the surface charge. “Nor can free Tht ions have much effect, for they disappear in the dialysis and yet leave the pure colloidal solution practically as active’ as before. The real agent must be the particles of colloidal thorium hydroxide which gather about them groups of H* ions and carry them into the surface in larger numbers than would be possible for the H* ions alone. The nature of this selective action must be connected with the shape of the surface, or, to put it in another way, a particle must reach a certain size before it can be regarded as liaving a surface-layer about it with a tension and an electric charge. We have at present in order of size—ions, ionic micelles (Prof. McBain, ‘Soap Solutions,” Nature, March 10, 1921), ultra-microscopic colloidal particles, micro- “scopic and macroscopic particles including gas-bubbles. At what stage a surface-layer is formed it is difficult to say, but it seems reasonable to suppose that the curvature of such a surface would have an effect on the charge adsorbed. The change of sign with decreasing size of air-sphere shown in these experiments seems to bear out this idea. The information obtained regarding the effect of thorium nitrate on the electrification of air-water surface layers may be summarized as follows :— 1. Thorium nitrate in aqueous solution and in concentra- tions as small as 8x10-® normal gives a positive electric charge to the surface of a sphere of air immersed in it. (In distilled water the charge is always negative.) 2. For concentrations in the neighbourhood of 6 x10 normal a sphere initially negative becomes gradually positive as the sphere diminishes in size. 3. Colloidal thorium hydrexide in small concentrations of the same order also gives a positive electric charge to a sphere of air immersed in it. | 4. Colloidal thorium hydroxide also exhibits the reversal of the sign of the charge with a decrease in the size of the bubble. 5. It is suggested that this reversal of sign is experimental evidence of a relation between the curvature of the surface and its adsorptive power. Lecture-Room Demonstration of Atomic Models. 395 The experiments are being continued as time permits in the hope of obtaining some new information regarding these free surfaces. Is is the intention to compare with thorium the effects of one or two other tetravalent and_ trivalent metals in the colloidal state. I wish to thank Professor J. C. McLennan for his kind and encouraging interest in the work. XXXIX. Note on a Lecture-Room Demonstration of Atomic Models. By Louts V. Kine, D.Sc., Macdonald Professor of Physics, McGill University *. | Plate IT. } Section 1. EVERAL mechanical models illustrating various types of atomic structure have been proposed from time to time. Among these we may mention Mayer’s classical experiments with floating and suspended magnets, illus- trating the action of atomic forces t. Many modifications of these classical experiments have been suggested. In particular, a paper by R. Ramsey de- scribes interesting modifications of the original apparatus f. Actual apparatus illustrating the supposed structure of atoms can now be obtained ready for use from scientific instrument makers §. All these methods involve the repulsive forces between steel elements (needles or spheres) in a permanent magnetic field, together with the central attraction set up by a per- manent magnet. An important point contributing to the success of the experiment is that all the magnets, repre- senting electrons, have as nearly as possible equal pole strengths. Owing to magnetic reluctance and effects of demagnetization, these conditions are difficult to realize in practice without a considerable amount of care and ex- penditure of time. * Communicated by the Author. ¢ J. J. Thomson, ‘ Corpuscular Theory of Matter’ (1907), Chapter 6, pages 105 et seq. t R. R. Ramsey, ‘The Kinetic Theory of the Electron Atom.” Pro- ceedings of the Indian Academy of Sciences, 1918. Phil. Mag. vol. xxxiii. Feb. 1917, pp. 207-211. § W.M. Welch, Scientific Company, Chicago. 396 Prof. L. V. King on a Lecture-Room Section 2. The magnetic elements which form the essential feature of the apparatus to be described consist of a number of steel spheres or small soft-iron rods magnetized in a strong alternating field. One evel model is shown diagrammatically in Pl. I. fig. 1, while fig. 2 shows the actual apparatus. The coil A consists of 340 turns of number 12B. «8. copper wire (2 mm. diam.); inside radius of winding 8°8 cm., outside radius 13°5 em., width of coil 3°9 em. Such a coil has a resistance of approximately 1°3 ohms and self-inductance of about 32 millihenries. It may be connected directly to a 110-volt 60-cycle A.C©. circuit without overheating. In such cir- cumstances it draws a current of about ¥ amperes. It is approximately of such dimensiors as to. give a maximum field strength at the centre of the coil. Placed over the opening of the coil is a large watch-glass B whose radius of curvature is approximately 25 cm. If available, an accurately ground concave glass mirror may be used to advantage. If, now, a supply of steel ball- bearings about 3 mm. in diameter is available, these may be placed on the concave surface B, where they will experience an attraction towards the lowest point approximately pro- portional to the distance. When the maximum current is passed through the magnetizing coii, the steel spheres will become A.C. “magnetic doublets of very uniform magnetic moments. It will be noticed that the magnetic axis will always be very accurately along the direction of the mag- netic field, independently of the rolling motion of the balls. Furthermore, if the spheres are of fairly uniform quality and the field strength sufficiently great, the instantaneous mag- netic moments of these doublets will be equal in magnitude and phase. In these circumstances the steel spheres will repel each other with a force varying as the inverse fourth power of the distance, the constant of proportionality being accurately the same for all the spheres. With the attraction to the centre varying as the distance, it may be expected that the magnetic elements will form remarkably symmetrical stable groupings. One such grouping is illustrated in Pea enos 3) (a). It is obvious that by a very simple arrangement of lenses and mirrors this model atom may be projected on a screen. The concave surface B may, if desired, be mounted so as to allow of rotation, thus increasing the interest of the “atomic ” arrangements. “This experiment is extremely convenient for lecture-room purposes, as it requires no preparation and is Demonstration of Atomic Models. hs ah _always certain to give results which never fail to delight an audience. An interesting variant of this experiment is to make use of the arrangement of two coils described in Section 5 (figs. 5 & 6). A surface of clean mercury is placed midway between the two coils. A number of steel balls floating on this surface will repel each other as already described, and will all tend towards the centre, owing to the greater intensity of field. The remarkably regular + arrangement taken up under these conditions is shown in ne 3 (b). The damping is so slight that the system may be set into oscillation in various ways by means of external magnets, giving a good illustration of internal vibrations in the atom. It would, moreover, be possible with no very great expenditure of labour to deter- mine the frequency of various modes and compare the results with theoretical calculations. Section 3. The same apparatus may also be used to illustrate the motion of the molecules of gas or the Brownian movements. For this purpose an elongated piece of iron is employed, e.g.a short cylinder of iron or steel wire about 1 em. in leneth by 1 mm. in diameter. In the alternating field of the coil such a magnet experiences a very strong torque, which vanishes when the axis lies along the direction ot the resultant A.C. field. If sucha magnet is placed in a flat eylindrical glass vessel occupying the centre of the coil, ai.d the field suddenly applied, violent movements of the little iron rod will be observed. The instantaneous moments set up by the field will be sufficient to make the rod leave the surface on which it is resting and describe a trajectory under the combined efféct of oravity and the magnetic field. At the termination of the flight, it will again strike the glass plate and will then receive an additional impulse made up of the magnetic torque and the elastic reaction at contact with the glass. This will start it on a new trajectory, and the process will be continued indefinitely until the rod makes contact with the plate at the termination of its flight in such a way that the instantaneous torque is zero. Then it stops dead with the axis pointing along the direction of the field. This is an event which happens very rarely. Several such rods enclosed within a glass vessel will keep in constant motion in a manner resembling the motion of molecules in a rarefied yas. An interesting variant of this experiment is to insert short steel wires along the diameters of small pith balls which hop around, describing flights in the glass vessel as if | 398 Prof. L. V. King on a Lecture- Room they were animated with life. As before, the glass vessel and its contents may be projected on a screen, the resulting effect being illustrative of molecular movements. Section 4.—Haperiments on Electrodynamic Repulsion. Owing to the distribution of the magnetic field around the coil employed j in chis experiment, the same apparatus is well suited to the demonstration of electrodynamie repulsion. For this purpose several plates of aluminium or copper should be cut with a radius approximately equal to the outer radius of the coil. Such a disk may be anchored by three strings fastened at equidistant points of the circumference so as to allow it to move vertically, with its centre over the axis of the coil, which is laid in a horizontal position. On applying A.C. circuit, the plate will float three or four centimetres above the coil. By placing a light iron rod (3 cm. x 1mm.) on the plate, the direction of the A.C. field is easily demonstrated, as shown in Pl. IT. fio. 4 It will be noticed that over an annular region bounded by the outer edge of the plate and a circle of half its radius, the lines of force are inclined at approximately 45° to the vertical. It is the reaction of the horizontal component of the A.C. field with the induced current due to the vertical component which causes the repulsion referredto. To demonstrate this, a circular plate may be cut up into several concentric rings and laid on a sheet of glass. When current is applied it is only the outer rings which are repelled, the force on the inner rings gradually becoming less, until that on the central. disk in a practically uniform field perpendicular to its plane is practically al. Tron filings poured on a glass plate laid horizontally over the coil assume an interesting laminar distribution, which again may be projected on a screen. The iron filings tend to arrange themselves in a series of vertical planes about iene high arranged radially. It is easily seen that this arrangement i is nee to the fact that under the influence of the alternating field, each of the radial planes represents a series of vertical A.C. magnets which repel each other. Their height is limited by the vertical stability of the plates under the combined effect of gravity and of the alternating field. Section 5.— Experimental Model of the Rutherford Atom. By using two coils of the dimensions already described, arranged with their planes horizontal at a distance apart equi al to the mean radius (Helmholtz arrangement), it is Demonstration of Atomic Models. 399 possible to secure a fairly uniform field over a considerable area midway between the coils. Such arrangement (Pl. If. figs. 5 & 6) allows of interesting experiments ona model atom approximating more closely ‘to modern ideas. A shallow circular basin of mercury is placed on an adjustable stand between the two coils. A number of steel pins with glass heads serve as the elements (electrons) for the model. It one of these is placed with the glass head on the mereury surface, it will float in a vertical “position and tend to move towards the centre ot the field, owing to the greater concen- tration of lines of force. This force towards the centre may be varied at will by adjusting the height of the mercury surface, or by placing rods of soft iron along the axis of the coils at adjustable distances above or below the mercury surface. Ifa second pin be floated on the mercury surface, it will repel the first with a force varying nearly as the inverse square law when the distance apart is not too great. A third pin may be added, when a triangular arrangm lent will be formed. Successive pins give the familiar series of regular polygons arranged in Senet nie rings. It is evident that the vreat advantage of the A.C. field is to make the mag- netical polarity of each of the pins very nearly equal, thus giving rise to a remarkable symmetry in the arrangements formed, as illustrated by figs. 7 (a) and 7(b} (Pl. IL). As before, the experiment can be carried out in such a way that the various stable arrangements may be projected on a screen. It isextremely simple to demonstrate the apparatus at a moment’s notice, the only precaution necessary being to use clean mercury so as to allow a great mobility of the floating pins on an uncontaminated surface. It is interesting to notice that rotation of the basin con- taining mercury does not disturb any particular stable arrangement, owing to the fact that the centrifugal force is accurately balanced by the change of slope of the para- boloidal mereury surface. The use of an A.C. field allows of the possibility of realizing positive electrons and a central nucleus, the law of forces between them being very nearly that of the inverse square and at the same time very exactly that corresponding to charges of te, +2e, +3¢, etc. It is evident from fig. 5 (PT: I1,), illust: rating the model under consideration, that ie ons may be represented by lengths of soft-iron wire of the same diameter arranged to move with both ends in the same plane at distances not too far apart compared with their length. In these circumstances we have repulsion according to the inverse square law, the charge —e being represented Cae ees TH Tb Senna ee i 400 Lecture-Room Demonstration of Atomic Models. by the average pole strength -+-m of each rod, which is extremely uniform. A nucleus of positive charge ne may be made up by taking 2n lengths of the same wire and inserting them in a small glass or aluminium tube, as shown in fig. 5, illustrating a nucleus of charge + 2e. In these circumstances, each of the rods representing electrons is attracted to the nucleus with a force varying nearly as the inverse square of the distance and proportional to nm xm, the average pole strength of each end of the rod being bo, In order to realize this arrangement, the rods (about 7 cm.x 1 mm. diameter), representing negative electrons, should be suspended from silk fibres about 1 metre or more in length. By adjusting the position of the rods in the space between the coils, a position of neutral equilibrium may be found in which there is practically no tendency for the rods to move either towards the centre or radially outwards. Under the combined effect of gravity and of the magnetic field they seem to float in any position. When this adjustment has been made, the rods representing the nucleus should be set in position along the axis of the coils. The suspended rod representing the electron may then be projected so as to describe a path about the fixed nucleus, and a damped elliptic orbit will be observed, the nucleus being at one focus. If 1 two lengths of. wire are ued to make up a nucleus +e in the manner illustrated by fig. 5 (a), we obtain a model of the hydrogen atom which is dynamically stable. If we make up positive nucleus of charge 2¢e, represented by two pairs of iron rods, we obtain a model (fig. 5 (d)) of the ionized helium atom which is dynamically stable. If we introduce an additional iron rod representing an electron (fig. 5), and therefore a complete helium atom, it seems impossible to obtain a dynamically stable arrangement by any circumstances of projection. For instance, any attempt to reproduce the symmetrical oscillation suggested by Langmuir meets with failure, owing to the dynamical instability cf this arrangement. It is obvious that further experiments along these lines, leading possibly to results of great interest, might be carried out by constructing large solenoidal coils to give a uniform A.C. field, in which circumstances the inverse square law of attraction and repalein between electrons and nuclear charges ne (n=1, 2, 3, etc.) would be faithfully reproduced. ee ee ee ae | 401] XL. The Influence of the Size of Colloid Particles upon the Adsorption of Electrolytes. By Humpnrey D. Murray, Exhibitioner of Christ Church, Ouford*. we, EVERAL workers have examined the influence of con- K- centration upon the coagulation of colloidal solutions, but references to the effect produced by alteration in the degree of dispersion are few and not very definite. Kruyt and Spek f examined the coagulation of colloidal arsenious sulphide, and found that the coagulative value of univalent ions increased with increasing dilution; in the case of a divalent ion there was a slight decrease ; whilst for a ter- valent ion there was a rapid decrease in the coagulative value. Burton and Bishop t examined the coagulative values of various ions upon colloidal solutions of arsenious sulphide, copper, and gum mastic, and as the result of their experiments found that with univalent ions the concentration of the ion required for coagulation increased with decreasing concentration of the colloid, for divalent ions the-concentra- tion of the ion was nearly constant, for trivalent ions the concentration of the ion varied almost directly with that of the colloid. More recently Weiser and Nicholas§ have extended these researches to colloidal solutions of hydrous chromic oxide, prussian blue, hydrous ferric oxide, and arsenious sulphide. They found in the case of the first three that the coagulative values of electrolytes tended to increase with dilution of the colloid, but the increase was less marked with electrolytes having univalent precipitating ions, and became more marked as the valency rose. Odén found that sols with ultramicroscopic particles are more sensitive to electrolytes than those containing amicrons. The object of these experiments was to examine the influence of the size and uniformity of colloid particles upon the adsorption of electrolytes as measured by the minimal concentration for coagulation. For this it was necessary to obtain solutions of the same colloid prepared under identical conditions, but containing particles of different mean size. It was decided to employ Odén’s method of fractional coagulation. The most suitable colloid to use, therefore, is one which, when first made, contains particles of markedly * Communicated by the Author. + Kruyt and Spek, Koll. Zeit. xxv. p. 1 (1919). t Burton and Bishop, Jour. Phys. Chem: xxiv. p. 703 (1920). § Weiser and Nicholas, Jour. Phys. Chem, xxv. 742 (1921). Phil. Mag. 8. 6. Vol. 44. No. 260. dug. 1922. 2D 402 Mr. H. D. Murray on Influence of Size of Colloid different size, and is stable when precipitated, redispersed, and dialysed. Gum mastic was found best to meet the requirements, and was used in the subsequent experiments. To show that the solutions employed were comparatively stable, the concentration of NaCl required to precipitate one — of the fractions at the beginning and end of the experiments was measured and found to be :— Bebwibth, ex i ae ae 433 millemols. Manig2Otle\) anaes, 439 ii Fractionation. One gram of finely-powdered picked gum mastic was dissolved in about 20 c.c. of alcohol, and poured slowly with vigorous stirring into one litre of distilled water. By this method seven litres of mastic solution were prepared. Odén recommends that in all cases NaCl should be used for the precipitation. With mastic this necessitates a very large concentration of salt, which appears to be strongly adsorbed, and comes slowly through the dialyser. It was thought better to employ HCl, which precipitates in smaller concen- tration. It was found convenient to separate the mastic into seven fractions with these concentrations of HC] :— Concentration of HCl Condition of Fraction. in millemols. Precipitate. Hae eS ooh as 0-11 trace 1 Brae Arcee 11-14 gooa 1 BE Dee 1°4-1°7 Vota wnat 1:7-2:0 - RE AA 2°0-2°3 » S\ i ley tae 2°3-2'6 small TA El De bh eee 2°6-2'°9 trace The procedure was as follows :—200 c.c. of the’ mastic solution were mixed with a quantity of 7 HCl to give the required concentration, and then poured into the centrifuge vessels and allowed to stand for 60 minutes from the moment of mixing. It was then centrifuged at 3000 r.p.m. for 30 minutes. At the end of this time the supernatant liquid was poured off and the precipitate carefully shaken up with about 100 c.c. of distilled water. I ractions II. and VI. were retained until about 1500 c.c. of each had accumulated; the other fractions were rejected. At the same time 1500 c.c. Particles upon the Adsorption of Klectrolytes. 403 of mastic were completely coagulated with a concentration of 3°0 millemols. of HCl, and redispersed i in an equal quantity of water. It appears below as solution B. It is to be expected that Fr. II. will contain particles of an average size greater than those in Fr, VI. and both will contain particles of more uniform size than those in solution B. | The solutions after dispersion were kept in dialysers of parchment paper until the dialysate was uncontaminated with HCl. They were then placed in perfectly clean vessels of resistance glass fitted with a siphon, and a soda-lime tube attached to the air-inlet. The siphon pipes were closed by short pieces of rubber tubing and pinch cocks. Basis of Comparison. Any method of comparison between two or more solutions based upon the total masses of the disperse phase in unit volume is useless when applied to data due to adsorption. It is possible to take as a basis the number of particles in unit volume, or, what is probably more characteristic and capable of giving more directly comparable results, the total inter- facial surface in unit volume. The former may 1n most cases be ascertained by a direct count under the ultramicroscope. To evaluate the latter it is necessary, beyond this, to know the total mass of the disperse phase, which can be effected by weighing after evaporation to dryness, or by the methods of volumetric analysis. In addition it demands a knowledge of the density of the disperse phase, or of the specific gravity of the solution and of the dispersion medium. Perrin in his researches upon Brownian movement. obtained the density of the mastic with which he was working by evaporating a portion of his suspension to dryness and estimating the density of the solid mastic. This value (1:064) he found to agree admirably with the density as determined from specific gravity measurements. It seems uncertain, however, as Burton * has pointed out, whether it is justifiable to assume that the density of the particles in the ordinary colloidal solution of gum mastic is the same as that of the solid substances. Perrin, as a matter of fact, used a suspension of mastic which had bean obtained by centrifuging the larger particles from a solution of mastic and rejecting the remainder. In the case of the present solution, it seemed desirable to determine the density of the particles directly, with a pyknometer. * Burton, ‘Physical Properties of Colloidal Solutions,’ 2nd Edition, p. 125 ah Dae: 404 Mr. H. D. Murray on Influence of Size of Colloid Concentration of Mastic. Thirty c.c. of the three solutions were evaporated slowly to dryness in a steam oven, and, as a mean of several deter- minations, gave the following weights of mastic in 10 c.c. of solution :— Weight found. SoluttvomGB: ere ncscseseseesee seer ‘00463 gms. GU egh ER Re ies ange tes cc sO Olas. Hine Le Ae cctee sero ne ta eeeee eet 00339 __,, Number of Particles. A true ultramicroscope was not used to count the particles, but a cardioid condenser, fitted to an ordinary microscope. The chief difficulty in work of this nature is to ascertain accurately the volume of the liquid within the field of view. A cell was made according to the recommendations of Siedentoft *, the only alteration made being the substitution of heavy glass for fused quartz. Fluorescence due to the glass was not sufficient to render difficult the counting of the comparatively large mastic particles. The cell consists of a glass plate, 5 cm. in diameter and 1:0 mm. in thickness, provided with a circular groove. The portion enclosed by the groove, 1 cm. in diameter, was polished exactly 2 pw ‘deeper than the surface of the plate. This was used with a cover slip about ‘25 mm. in thickness. The cell was soaked in concentrated sulphuric and chromic acids, washed with water, and then passed through two solutions of re-distilled alcohol. It was finally famed. The source of illumination was a Pointolight lamp, fitted with a condenser. All the solutions examined were diluted with water which had been carefully distilled and allowed to stand for a month undis- turbed. It contained on an average 1 particle in 20 counts in a volume of 14°1 x 107° cu. mm. and could, there-. fore, be considered optically pure to the degree of accuracy to which work was carried. All the solutions were contained in vessels of resistance glass, closed with corks covered with tinfoil. The method of procedure was to transfer, by means of a clean platinum loop, a very small drop of the solution to be examined to the central portion of the cell. The cover slip was laid on and pressed down until the Newton interference rings appeared at the edges. The dilutions were such that, when viewed with a convenient stop in the eyepiece, about three or four particles appeared * Siedentoff, Verhd. Deut. Phys. Ges. xii. p. 6 (1910). Particles upon the Adsorption of Electrolytes. 405 in the field of view. One hundred counts were taken at half- minute intervals, and the average number deduced from this. A few of the particles, especially in the case of Ir. II., tended to adhere to the walls of the cell, and to prevent any error due to this, the field of view was shifted five times during each count. The results obtained were as follows :— une | iT)ie } | if Solution. Dilution. . Obe Hye- aa o Volume of Field oes ‘ ‘| jective. | piece. | _. of view. ns | Vlew. Particles, |-- _- SC es |Soln.B..) 726 |4mm.f.]. x12 -30 mm.| 14:1«10-° mm.? 41 IFr. II... 396 aioe) AD). (SO 141% 10-5 . 3:0 i | | iFr.VL..) x3896 | 43 ee ee LS ee eae xX LOT 5 4-0 ee i | Density of the Particles. The density of the solutions was determined with an accurate pyknometer in a thermostat at 17:2°C. The weighings agreed to ‘0002 gm. Fr. VI. was too dilute to give accurate results. The specific gravity of each solution rose slightly during dialysis, owing probably to the removal of adsorbed or dissolved alcohol. ‘This rise continued for about five days. The weighings were made at the end of ten days. The dialysis was then continued in more efficient dialvsers made by Soxhlet thimbles impregnated with collodion, but the specific gravity remained constant. Asa mean of four weighings for each solution, the following values for the density of the particles were obtained :— Holm. gh:— 14195. Fe MY3=1'186: As a mean the density of the mastic was taken to be 1°190. This value is considerably higher than the density of the mastic in bulk, owing possibly to changes occurring either on dispersion, or coagulation. Perrin* states that he observed the densitv of his carefully washed granules apparently to rise in salt solutions, and this may account in part for the difference. Size of Particles. From these three sets of data—the number of particies in unit volume, the total mass of mastic in unit volume, and the density of the particles—it is possible to calculate the mean * Perrin, Ann. Chim. Phys. xviii. p. 5 (1909). 406 Mr. H. D. Murray on Influence of Size of Colloid radius 7 of the particles in each liquid. Of these three measurements that of the density seemed possibly least accurate, but, as it occurs in each calculation, the relative sizes remain unchanged. Pe ‘Total Mass; Mean volume Radius . Solution.) -~ "3 ‘jof Particles of one CK (72 LOL VT i hiss (7s | ap in 10\cie,\), * particle. Soln.B..| 211 10° | 0046 gm. | 1:83x10-7 yp? | 164 wp 564 Hr oEL. 842 <0 0034 eas Oe ea 2 Oli 340 | By, V1. |.|-252 | "S 19:00 millemols./litre Slight settling Clear with large flocks Clear = We 22 . | Clear with large flocks % ‘: L Aa 114 ” ” | ” ” ” 20 millemols, | >20 millemols. 9) Srerizs ITT, Rate of Coagulation of Mastie with Al,(SO,)3. 15 minutes. 60 minutes. 5 hours. 15 hours. 10 9) >20 3 1:04 millemols, >20 ‘5 >20 ry, ‘23 Series LV. Rate of Coagulation of Mastic with NaCl. 38 millemols. ) 25 hours. °38 millemols. "15 " 48 hours. °38 millemols. Time after Mixing ..........0.00 30 minutes. 65 minutes. - 2 hours, Liminal Concentration of NaCl. Dilution 100 p, c./Ca, 1500 millemols. 2) 10 ? 630 millemols.! 450 millemols, >1500 SIiG00) > 3, 900, 810 5 hours. 450 millemols, 9? 24 hours. 360 millemols. 720s, , 48 hours. bars 360 millemols. 720 3 410 Mr. H. D. Murray on Influence of Size of Colloid two vessels, one slightly smaller than the other, and fitting by a ground-in joint inside the larger. The smaller vessel has a slightly higher inner cylindrical vessel, the base of which is concentric with that of the outer vessel and fused to it. The electrolyte solution is placed in the inner vessel and the colloidal solution in the annular space, both having been previously rinsed out with their respective solutions. The larger vessel is placed over the smaller, and the whole inverted and left for 30 seconds millemols oncentralion of NaCl in L = Sa 2k 36 48 Time in hours: after mixing to drain. By this means a sudden and complete mixing of the two solutions is obtained: The mixed solution is then poured into a vessel of hard glass and corked. The whole apparatus, as were all the vessels with which the mastic solution came in contact, is made of hard glass and was steamed out between each series. The experiments were conducted with 10 c.c. of the mastic solution at the required dilution. Into the inner vessel was poured enough water to make the volume of the elec- trolyte solution up to 9 c.c,, and then the latter solution was added at a convenient concentration. To determine Particles upon the Adsorption of Electrolytes. 411 one end point for a given concentration of mastic anda given electrolyte, four solutions were made up with a fairly wide difference of concentration in each solution, so-as to give a large bracket. After standing 48 hours and centrifuging for half an hour at 2000 r.p.m., at which speed there was no sedimentation of the pure mastic solution, four more solutions were made up, in which the concentrations of electrolyte were such as to cover the interval between the r ra = yo in millemols, Concentration of Al,(S0,), Time in hours after mixing two concentrations in the first determination, within the limits of which the end peint was observed to lie. The process was repeated until the limit of observation was reached, and eventually gave two concentrations of which it was possible to say that one definitely caused complete coagulation, and one did not. The end point was taken as the mean of these two concentrations. The observation of the solutions was made by daylight against a black back- ground. The size of the final bracket of concentration varied directly with the concentration of electrolyte necessary for coagulation, and the results were therefore more accurate with trivalent ions than with monovalent. The results were as follows (Series 5, 6 and 7, and figs. 3, 4 and 5). IP 19 G8 GOT S6l Srl PIT FOG ay oe Sua ie GP 89 16 SLT 96T 61 TSI 96 ee SUNG eee) ee GL éII IGT 88I yGG> te | uve 106 OLE ee edie Oraeen| cordgz, | 9 dion | a dees, | od Oy | od 1.97 | 0 '°d¢.gg | ‘0d 1.99 Josey Jo WOlZerjUe0 KOH [Truly ‘OOvTING [VlovyiopUy e of Colloid ~ ~ a 06L 069 Ole | Sane 06h 09 TA“) -sromertuu af (Oey 0&9 084 OIG O&P ee OOP Je eu veaquaou0/ r | 009 OFe O1¢ OIF a OF are -uqog | JO UOr}eVAZUBOUOH [TVULUNTD ‘a 'd 0% od 7.94 | ° ‘d ¢.e¢ Od .O7 2 “d @.eG ‘ad 1.99 ‘OLISVI JO UOTFVAJUEDMOH [eULT TORN WUL[NSvoy —]] A saTUAG Mr. H. D. Murray on Influence of S&: | au 16 re aes STAM] gorau wt "igva 8.9 @.9 6 0.6 Hees cer cutog JO u01je.1yUI0U0D eure ‘od ET "ad 1.97 | ‘od OF ‘od €.6G ‘a “d 1.99 ‘OIISV] JO UOIFVAPUOOUOD [VUTY | Vg yuRRoBVo)—'] A SUIYAS a CFI- GLI Wa ee a. ee occas are es : Oi | Ob. (One Woe lees | Se re ra) ee ee OLT. COG. cee in wo Li Sa a ‘ujog J Lyey O) [LAE a ‘od get | o-doz |) 'd 49g | 9 'deeg | ‘o-dop 0 'd¢g.eg | od 1.99 “OIISLIW[ JO WOTyVAzUeOUOD [BULA 412 FOO )IV que|n.ovory— A SHTYAS of A1,(S0,), in @illemols Cofitentration Concentration of BaCt, in millemols, tM || FEF Concentration of NaCl in millemols. 414 Size of Colloid Particles and Adsorption of Electrolytes. Discussion of Results. It will be seen that, under the conditions imposed and within the limits of the experiments, a comparison of the . data obtained, upon the basis of the total interfacial surface _ in unit volume, leads to uniformity in the curves. Such uniformity is not to be observed when the comparison is based upon the mass of the disperse phase, or the number of particles, in unit volume. It appears that adsorption is very largely conditioned by the amount of interfacial surface exposed. It is to be noticed, however, that the minimal concentration of electrolyte is higher throughout for the fraction containing small particles than for that containing large particles. This may be brought about in two ways. The smaller particles may bear a higher charge per unit area of their surface, or the critical value to which their charge must be reduced before coagulation begins may be lower than in the case of the larger particles. The latter explanation is more probably correct, since it is known that the surface tension of larye particles is greater than that of small ones. It seems probable, if the existence of a critical potential difference for coagulation between a particle and the dispersion medium be admitted, that this should be lower in the case of small particles which have Jess tendency to adhere, and should thus permit of a greater freedom of approach between the particles. If the former explanation were correct, we should expect a separation of the particles according to size in an electric field; but this is contrary to experience, the particles move at the same rate independently of their size. According to the Helmholtz theory of the electrical double layer, this effect is due to equal density of the charge upon unit area of the surface. It appears probable, therefore, that the smaller particles have a lower critical potential difference for coagulation. The behaviour of the solution containing mixed particles of different size is in some respects curious. With both Al,(SQs)3 and NaCl the curve representing the coagulation of this solution is more flattened relatively than the other two curves. A lack of uniformity in the size of the particles appears to render the solution less sensitive to change in concentration, in the case of coagulation by univalent and trivalent ions. Notices respecting New Books. 415 Summary. (a) A separation of the particles present in a suspension of gum mastic has been effected by Odén’s method of Ee actional coagulation. (b) The density of che particles, and the mass of mastic and the number of particles in unit volume have been measured, and from them the interfacial surface in unit Salanie calculated. (c) The variation of the minimal concentrations of Al,(SO,4)3, BaCl,, and NaCl to coagulate solutions containing particles of different mean size with change in concentration of the solutions has been inv esti gated. (d) It has heen shown that uniformity in comparison of the results can be obtained upon the basis of the interfacial surface in unit volume. Ithas also been shown that, upon this basis of comparison, small particles require a higher minimal concentration of electrolyte than large particles. In conclusion I should like to thank Dr. A. 8S. Russell for his valuable advice and assistance. and Mr. H. M. Carleton tor kindly putting at my disposal the microscopical apparatus required. Christ Church Laboratory, Oxford, May 15th, 1922 XLI. Notices respecting New Books. Basic Slags and Rock Phosphates. By G. Scorr RopeErtson. Pp. xiv+112, 8 plates. 1922. Cambridge Agric. Monographs. Cambridge University Press. 14s. net. Os value of sciertific investigation of the results accruing from the use of phosphatic dressings on crop-production is obvious to all, but it gains in emphasis when, as Sir E. J. Russell points out in a preface to the above book, agriculturists have to realize that the composition of basic slag has undergone much change in consequence of the enforced modifications in the processes of “steel manufacture. We would go farther than Sir E. J. Russell and 416. Notices respecting New Books. say that even if the war had not given an impetus to the change over from the basic Bessemer and acid open-hearth processes, economic considerations would none the less have demanded the development of the basic open-hearth production of steel from low-grade iron-ores. ‘This result” (to quote from the preface) ‘“‘is, of course, distinctly awkward for the agriculturist who sees a valuable fertilizer disappearing, and being replaced by one which is more costly and at first sight seems to be nothing like as ood.” r After a review of the various scattered experiments on the use of rock phosphates and basic slags hitherto undertaken, Dr. Scott Robertson describes in detail the Essex experiments carried out in the winters of 1915, 1916, 1918, and 1919 under the auspices of the Hast Anglian Institute of Agriculture. The soils treated were those of the Chalk, London Clay, and Boulder Clay, and varied considerably in mechanical and chemical composition. The yields of hay and clover were correlated with the rainfall, and it was found that the drier the season, the greater was the increase in production due to the use of phosphates. The botanical results are also given, the crowding-out of the weeds and the covering of bare areas with grass being noteworthy. Dr. Robert- son’s main conclusion is that for root crops and late harvests with high rainfall, rock phosphates will prove a suitable substitute for the high-grade Bessemer basic slags. The careful records and correlations were made personally by Dr. Robertson at consider- able inconvenience and discomfort, and under most difficult circumstances. They are therefore the more valuable, and do him the greater credit. The latter part of the book is concerned with investigations of the large yields resulting from the use of basic phosphates. From botanical analyses it is evident that the open-hearth fluor-spar slags of low solubility are less effective than the non-fluor-bearing and therefore more highly soluble slags. The effects of the temperature and texture of the soil on the accumulation of - nitrates, on the soil bacteria, and on the acidity and lime-require- ment are clearly expounded, and the deductions emphasized by means of abundant statistics. Altogether, the work constitutes a most valuable contribution to agricultural knowledge. It is a pity that the publishers cannot retail this book of 112 pages and 8 plates for less than 14s. P. G. ae: Phil. Mag. Ser. 6, Vol. 44, Pl. II. lig. 7 (a). Fre. 7 (8). INING. Iie, 2. I'tg. 6. +m -m -m Fie. 3 (a), fap) (80) (an) ' a) ay gy ay eS as ad an an ) ) (av) (8) Fre. Tie. 3 (0). Fie. 5 +2m +m -2m +2m -m g e 3 3 -2m +2m +m -2m Lia +2m -em +m Phil. Mag. Ser. 6, Vol. 44, Pl. II. ia. 7 (a). Pia. 7 (0). ‘ rare mini a oer : { Pore ay? Tita ee LONDON, KDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. \ [SIXTH SERIES‘. S —_ St ate SAP TM Bie Bi ag22. XLII. The Disintegration of Elements by « Particles. By Sir E. Rourgserrorp, £.4.S., Cavendish Professor of Experimental Physics, and J. CHapwick, Ph.D., Clerk Maxwell Scholar, University of Cambridge *. 7 a former papert we have shown that long-range particles, which can be detected by their scintillations on a zine-sulphide screen, are liberated from. the elements boron, nitrogen, fluorine, sodium, aluminium, and_phos- phorus under the bombardment of « rays. The range ol these particles in air was greater than that of free hydrogen nuclei set in motion by @ particles. Using radium C as a source of a rays, the range of the particles varied from 40 cm. for nitrogen to 90 em. for aluminium, while the range of free hydrogen nuclei under similar conditions was about 29 cm. Previous experiments { by one of us had indicated that the long-range particles from nitrogen were deflected in a magnetic field to the extent to be expected if they were swift hydrogen nuclei ejected from the nitrogen nucleus by the impinging « particle. The nature of the particles from the other five elements was not tested, but it seemed very probable that the particles were in all cases H nuclei which were released at different speeds depending on the nature of * Communicated by the Authors. - + Rutherford and Chadwick, Phil. Mag. vol. xlii. p. 809 (1921). t Rutherford, Bakerian Lecture, Proc. Roy. Soc. A, vol. xevii. p.374 (1920). Phil. Mag. Ser. 6. Vol. 44. No. 261. Sept. 1922. 2E 418 Sir E. Rutherford and Dr. J. Chadwick on the the element and on the velocity of the incident a particle. Under the conditions of the experiment, these H nuclei could only arise from a disruption of the atomic nucleus by the action of the « particles. | : Attention was also drawn to the remarkable fact that in the case of the one element examined, viz. aluminium, the particles were liberated in all directions relative to the incident « particles. | In the present paper we Shel give an account of ex- periments to throw further light on these points and to test whether any evidence of artificial disintegration can be observed in the case of other light elements. Magnetic Deflexion of the Particles. In the course of this work, the microscope used for the counting of scintillations has been further improved. For the present experiments it was essential, in order to obtain a sufficient number of scintillations per minute, that the area of zinc-sulphide screen under observation should be greatly increased without diminution of the light-gathering power of the microscope system. Following the suggestion of Dr. Hartridge, a modified form of Kellner eyepiece was constructed. A planoconvex lens of about 7 cm. focal length was placed so as to render the rays of light from the objective approximately parallel, and the image so formed was viewed through an eyepiece consisting of a similar lens and an eye-lens of 4 cm. focal length. Used in conjunction with the old objective, Watson’s Holoscopic of 16 mm. focal length and *45 numerical aperture, this system gave a field of view of a little more than 6 mm. diameter. A rectangular diaphragm was placed in the eyepiece, limiting the field of view to an area 6 mm.x4°9 mm. Our previous system had a field of view of 8-3 sq. mm. area, so that the new microscope, under similar conditions, gave about three times the number of scintillations of the old, The precautions adopted in counting were similar to those described in our previous paper. | The method of measuring the magnetic deflexion of the long particles was very similar to that described by one* of us in the Bakerian Lecture of 1920. The pa arrangement is shown in fig. 1. The source of & rays was placed at R and was inclined at an angle of 20° to the horizontal. The lower edge was level with the face of a brass plate S which acted as a slit, * Rutherford, loc, eit. Disintegration of Elements by « Particles. 419 The distance from the centre of the source to the farther edge of the slit was 2°95 cm. The carrier of the source and slit was placed in a rectangular brass box between the poles of an electromagnet, the field being perpendicular to the length of the slit. A current of dry oxygen was circulated through the box during the experiment. Pie A An extension piece L, projecting 1°7 cm. beyond the edge of the slit, was fixed to the carrier in order to increase the amount of deflexion of the particles issuing from the slit. In the end of the box was a hole 1 cm. wide and 2 cm. long covered with a sheet of mica of 3°62 cm. stopping-power. The ZnS screen was fixed on the face of the box, leaving a slot of 1 mm. depth in which absorbing screens could be inserted. The source R was a brass disk of 1°2 cm. diameter coated with the active deposit of radium. Its initial y-ray activity was usually equivalent to about 40 mgm. Ra. The material, the particles from which were to be investi- gated, was laid directly on the source if in the form of foil, or if in the form of powder dusted over its face. The experiment consists in obtaining an estimate of the deflexion of the particles falling on the screen by observing the effect of a magnetic field on the number of scintillations near the line HE, the edge of the undeflected beam of particles. The position of the microscope was fixed in the following way :—After placing the source of « rays in position, hydrogen was passed through the box. The « rays could then strike the ZnS screen, and the edge EK of the beam was ciearly defined. The microscope was adjusted so that the 2H 2 420 Sir E. Rutherford and Dr. J. Chadwick on the edge of the beam of scintillations appeared a little above a horizontal cross-wire in the eyepiece of the microscope, marking the centre of the field of view. When the magnetic field was applied in such a way as to bend the & particles upwards (called the positive direction of the field), the edge of the beam is deflected downwards in the field of the microscope and the scintillations appear only in the lower half. When the field was applied in the opposite direction (negative field), the edge of the heam moved upwards in the field of view. The strength of the magnetic fields used in the experiments was always such that the whole field of view was covered with scintillations when the negative magnetic field was applied. In the experiments on the magnetic deflexion of the long-range particles, the number of particles is far too small to give a band of scintil- lations with a definite edge. It is clear, however, that if the particles are positively charged, the number of scintil- lations observed with the negative magnetic field will be greater than the number observed with the positive field, and that the ratio of these numbers will give a measure of the amount of deflexion of the particles. By determining this ratio for the long-range particles and comparing it with that for projected H particles of known velocity, we can obtain an approximate value for the magnetic deflexion of the long-range particles. The general method of the reduction of the observations is perhaps best shown by an account of the experiments on the particles from aluminium. EKaperiments on Particles from Alumimum. After fixing the position of the microscope in the way described above, an aluminium foil of 3°37 cm. stopping- power was placed over the source. Dry oxygen was passed through the box, and a mica sheet of 10 cm. stopping-power was inserted in front of the ZnS screen. The total absorption between the source and screen was then equivalent to 30cm. of air. The scintillations observed were consequently due to long-range particles from the bombarded aluminium; the ranges of the particles under observation varied from 30 cm. to 90 em., the average range being about 45 cm. Counts of the numbers of scintillations observed with positive and negative fields due to an exciting current of 6 amps. were then made. The mean ratio of the numbers with a — field to those with a + field obtained from several experiments was 3°7. The observations were repeated with a1 a Disintegration of Hlements by « Particles. 421 a field due to an exciting current of 4 amps.; the corre- sponding ratio was 2°1. When the source had decayed to a small fraction of its initial value, the aluminium foil over the source was removed and a thin sheet of paraffin wax put in its place. The mica sheet in front of the ZnS screen was replaced by a sheet of 34 cm. stopping-power, making the total absorption equi- valent to 16 cm. of air. The scintillations observed on the screen were now due to H particles ejected from the paraffin wax of ranges between 16 cm. and 29 cm., the average range being about 22cm. ‘The ratio of thenumbers of scintillations for — and + fields was determined for an exciting current of 4 amps. and found to be 3:2. It appears from these results that the long-range particles from aluminium of average range 45 cm. were less deflected by the same magnetic field than H particles of average range 22 cm.; and that in the magnetic field due to a current of 6 amps., which was 1°34 times the intensity of the field due to 4amps., they were more deflected than were the H particles in the latter field. To a first approximation we may say that the value of mv/e tor particles from aluminium of range 45 cm. is 1°23 times greater than that for H particles of range 22 cm. This result is clearly consistent with the view that the particles from aluminium are H nuclei moving with high velocity ; for, assuming that the range of the H particle is pro- portional to the cube of its velocity, the velocity of a particle of range 45 cm. is 1°27 times that of a particle of 22 cm. range. These experiments show, therefore, that the particles from aluminium carry a positive charge and are deflected in a magnetic field to the degree to be anticipated if they are hydrogen nuclei moving with a velocity estimated from their range. While there can be little doubt that the particles are hydrogen nuclei, it is very difficult to prove this point definitely without an actual determination of the velocity and value of e/m of the particles. Our knowledge of the relation between the range and velocity of complex charged particles is too indefinite for purposes of calculation. On the other hand, if we assume, as seems a priori probable, that the ejected particle is the free nucleus of an atom, it is possible to show with some confidence that only a particle of mass 1 and charge 1 can fit the experimental results. Additional evidence as to the value of mv/e of the particles from aluminium was obtained by comparing their magnetic deflexion with that of the « particles of &°6 cm. range emitted by thorium C. In this experiment the source R was a very 422 Sir ®. Ratherford and Dr. J. Chadwick on the weak source of the thorium active deposit obtained by ex- posing a disk to thorium emanation. Hydrogen was passed through the box, and sufficient absorbing screens were inserted in front of the ZnS screen to cut out the 5 cm. a particles of thorium C. The numbers of a particles falling on the screen for — and + fields due toan exciting current of 6 amps. were counted, and the ratio of these numbers was found to be 2°4. Comparing this ratio with those found for the long-range particles, we see that the value of mv/e for the latter is about 0°8 of that for « particles of 8°6 cm. range, 2. e. about 3°4x10° e.m. units. The calculated value, assuming that the particles are H nuclei and that their velocity is pro- portional to the cube root of the range, is about 3°7 x 10°,e.m. units. Considering the difficulty of the experiments, the agreement is satisfactory. Haperiments on Phosphorus and Fluorine. Measurements similar to the above have also been made on phosphorus and fluorine. In the case of phosphorus, a thin layer of red phosphorus was dusted over the face of the source. The total absorption in the path of the particles was about 35 cm.; the range of the particles under observation varied therefore from 35 cm. to the maximum range of 65 cm., the average being about 45cm. The ratio of the numbers of scintillations for — and + fields due to current of 4 amps. was 2:0. In the case of fluorine finely powdered calcium fluoride was dusted over the source. Previous experiments have shown that no long-range particles are emitted fromcalcium. The total absorption in the path of the particles was about 30 em. The maximum range of the particles from fluorine is approxi- mately 65 cm., and the average range of the particles falling on the screen was around 40 cm. The ratio of the numbers of particles observed for — and + fields due to an exciting current of 4 amperes was 2°95. It is clear from these results that, within the error of experiment, the particles liberated from phosphorus and from fluorine are bent in a magnetic field to approximately the same extent as the particles from aluminium. We may conclude, therefore, that these particles also are H nuclei moving with high speed. We have not examined the particles from boron and sodium in this way, but there seems no reason to doubt that they also consist of H nuclei. Disintegration of Elements by « Particles. 423 The Ranges of the H Particles. In the experiments described in our previous paper only two elements, nitrogen and aluminium, were investigated in any detail. The other elements were examined in a qualitative manner, but it was shown that the ranges of the liberated H particles were in every case greater than 40 cm. of air. The ranges of the particles from these elements—viz. boron, fluorine, sodium, and phosphorus— have now been determined more accurately. Attention has been drawn to the remarkable fact that the H particles liberated from aluminium appeared not only in the direction of the incident « particles but also in the reverse direction. The number of particles emitted in the backward direction was of the same order of magnitude as for the forward, but the maximum range in the backward direction was smaller, being 67 cm. as against the 90 cm. range of the forward particles, for « particles of 7 cm. range. Some experiments with nitrogen showed that the number of H particles emitted in the backward direction was very small at absorptions of more than 18 ecm. of air. We have repeated these experiments and extended them to include the other elements boron, fluorine, sodium, and phosphorus, with the result that we find that in every case the H particles emitted on disintegration of the nucleus escape in all directions, the maximum range in the backward being less than in the forward direction. Fig. 2. I pea The experimental arrangement for the measurement of the ranges of the forward particles was the same as that described in our previous paper. The apparatus used in the investigation of the particles in the reverse direction differed from this in the arrangement of the source, and is shown in the diagram (fig. 2). The source of a particles was carried on a rod passing with a sliding fit through a stopper which fitted tightly into the 494 Sir E. Rutherford and Dr. J. Gitar on the brass tube T of-3 cm. diameter. The end of this tube was provided with a hole 7°5 mm. in diameter, closed by a silver Foil of 3°75 em. air equivalent. The zinc sulphide screen S was fixed on the face of the vessel leaving a slot in which absorbing screens could be inserted. The apparatus was placed between the poles of an electromagnet to reduce the luminosity produced in the screen by the B rays. The source R was a silver foil of 4:15 cm. stopping-power coated on one side only with the active deposit of radium. Its initial y-ray activity was in most experiments equiva- lent to about 30 mg. Ra. The inactive side of the silver foil faced towards the ZnS screen. The distance of the source from the screen was generally about 3°5 cm., but could be varied, and its position read off on ascale. The elements to be examined could in most cases only be obtained in the combined state. The powdered compound was heated in vacuo, and a film prepared by dusting on to a foil smeared with alcohol. The screen thus prepared was placed immedi- ately behind the source. As in our previous experiments, a stream of dry oxygen was circulated through the apparatus. In all cases, except that of nitrogen, the maximum range of the particles emitted in the backward direction was greater than the range of free hydrogen particles, so that no complication arises from the presence of hydrogen in the silver foils or other materials in the path of the @ particles. In the case of nitrogen, however, as our previous experi- ments had shown, the range of the backward particles is much less than that of free hydrogen particles, and it was consequently necessary to allow for the “ natural” effect, 7. e. for the H particles arising from hydrogen contamination of the source and screens in the path of the a rays. It was found inconvenient to use gaseous nitrogen for these experi- ments, and a suitable screen was prepared by sifting a thin layer of powdered paracyanogen, C,N,, on to a gold foil. The scintillations observed on the ZnS screen when the film of paracyanogen was placed against the source were due to the “natural” particles from the source and screens, together with those which came from the nitrogen in the paracyanogen. On taking away the film of C,N, the natural particles alone were counted. In some experiments a film of paraffin wax was placed against the source. The natural effect remained the same, showing that even if the film of paracyanogen contained a large amount of hydrogen the number of free H particles scattered to. the ZnS screen by the walls of the vessel was negligible. Figure 3 shows the type of results obtained in these Number per Milgram. Disintegration of Elements by « Particles. 425 experiments. The ordinates represent the number of scintil- lations observed per minute per milligram of activity of radium C, measured by y rays; the abscissze, the stopping power for a rays of the absorbing screens, expressed in terms of centimetres of air. The dotted curve A gives the natural effect observed when the screen of paracyanogen was absent or replaced by a film of paraffin wax ; the full curve B the effect when present. The difference of these curves there- fore represents the effect due to the nitrogen in the para- cyanogen. It will be seen that the maximum range of the Fig. 3. Forward and Backward Particles trom Mitrogen. Absorption in cms. of air backward particles from nitrogen is about 15 cm. Curve C is the absorption curve for the particles emitted by nitrogen in the forward direction. | In the following table are given the maximum ranges of the particles liberated from the elements which show the disintegration effect, for both forward and backward directions. — Element. Forward range. Backward range. Cr: cm, Borer Se Rates « 58 38 Brian 7 Le Aree 40 18 Mhrarme 1253)! 65 48 ecu Me BU lw et, 58 36 Alaremiony 92:22. 28.0.0 90 67 Phospherwua? 2:05. Mol 25: 65 49 426 Sir E. Rutherford and Dr. J. Chadwick on the _ It should be pointed out that the ranges of the forward particles from boron, fluorine, sodium, and phosphorus may be subject to considerable error, owing to the use of a film of powder as the bombarded material. The particles of maximum range are produced on the surface of the grains of powder, and theretore to find the true range the size and air equivalent of the grains of powder must be known. For the ranges given above it has been assumed. that the grains were uniform in size and an average value of the air equivalent of the film of powder has been calculated from its weight per sq.cm. ‘The ranges so determined are obviously somewhat less than the true ranges. The ranges of the backward particles are, of course, not subject to this source of error. It was observed that the number of particles liberated from the different elements appeared all to be of the same order of magnitude when allowance is made for the differ- ence in range. In our original experiments we found that the number of particles from boron was somewhat smaller than the numbers from the other elements, but this was due to the use of an irregular film. Using a film of more finely powdered boron it was found that the number of particles from boron was about the same as from the cther | elements. Haamination of other Elements. In our former experiments we examined all the light elements, with the exception of the rare gases, as far as calcium. Of these only the six elements of the above table were found to emit H particles in detectable amount under the bombardment of ¢ rays. As was pointed out in that paper, the atomic masses of these elements can be represented by 4n+a where n is a whole number, a result which receives a simple explanation on the assumption that the. nuclei of these elements are composed of helium nuclei of mass 4 and hydrogen nuclei. On the other hand, some of the light elements which gave no detectable number of H particles also had atomic masses given by 4n+a. It was thus a point of great importance to repeat the examination of these elements with the improved microscope, and to search, if possible, for the emission of particles of shorter range than free H nuclei. In some cases it was only possible, on account of hydrogen contamination of the materials, to observe at absorptions yreater than 30 cm. of air, while in others the observations were carried well within this range. Lithium was examined as oxide and as metal, a thin sheet of the latter being obtained by pressing molten lithium Disintegration of Elements by « Particles. A427 between two steel plates in an atmosphere of carbon dioxide. No evidence was found of any particles of range greater than 30 cm. Owing to hydrogen contamination of the Li and Li,O the observations at smaller ranges were not decisive. Observations in the backward direction revealed no detectable number of particles of range greater than 14 cm. Beryllium was examined as the powdered oxide, and there was again no evidence of the emission of particles of longer range than 30 em. in the forward direction or 15 cm. in the backward. Magnesium was examined with a sheet of the metal and also with a screen of powdered magnesium. There was no evidence of long-range particles. For silicon a screen of powdered silicon and a thin sheet of quartz were used. With the sheet of quartz it was possible to make observations in the forward direction at absorptions as low as 17 em. The scintillations observed were due entirely to the natural H' particles. Fig. 4, To Pump. Chlorine had been previously examined in the form of various chlorides. These observations were repeated, and the results confirmed the conclusion that particles of greater range than 30 cm. were not liberated in any detectable amount. In order to pursue the observations within the range of free H particles a special series of experiments was 428 Sir E. Rutherford and Dr. J. Chadwick on the carried out. A glass apparatus, similar in design to the standard ae) was used. The details will be clear from the diagram (fig. 4 In order to oid ‘the bombardment of the glass walla and consequent liberation of H particles the inside of the tube was lined with platinum foil. The surfaces of the brass plate B and of the rod carrying the source were protected from the action of the chlorine by a coating of hard pitch. The stopcocks and ground-joint were lubricated with a brominated grease. The source of «rays wasa platinum foil coated with radium active deposit. Pure dry chlorine was prepared by heating gold chloride, AuCl;, contained in the tube A, and was passed over P.O; before entering the vessel T. As an additional precaution a little P 205, was placed in the vessel itself. When the source was placed in position the air was removed by pumping and washing with dry carbon dioxide. Carbon dioxide was then let in to atmospheric pressure and the natural H particles were counted at absorptions varying from 16 cm. to 30 cm.- The carbon dioxide was then replaced by chlorine, and tbs scintillations at similar absorp- tions were observed. The chlorine was then allowed to be reabsorbed by the gold chloride and carbon dioxide let in again. In this way counts on the chlorine were included between counts of the natural particles, and any traces of adventitious hydrogen could be allowed for. The results showed no evidence of the liberation of H particles from chlorine in the range examined, 7. e. at absorptions more hes 1oxcm. oF alr, Discussion of Results. For convenience of discussion the atomic numbers and the masses of the isotopes of the elements from hydrogen to potassium are given in the following table. Of these elements aluminium is the only one which has not yet been examined for isotopes, but it appears likely that it is a pure element of atomic mass 27. With the exception of helium, neon, and argon, all the elements in the table have been tested to see whether H nuclei are ejected by the action of a particles. The six active elements, as they may be termed for convenience, are underlined. sintegration of Elements by « Particles. 429 TABLE J; , Atomic Atomic iy ; Atomic Atomi Element. Number. Masses. Klement. N eae Mikes, fe cn. tte [LL 1 POS IN Oe 1) 23 | —_— 1 Sg Sn 2 4-00 | be 12 24,25, 26 | 2 Sees 3 6,7 1 gece lea el oe 13 27 ae ee 4 9 lick iabradaal 14 28. 29 “ geliaat NEUE. NORA sable 15 31 7 edema st bedi: ee AL 16 32 N Packet ox... 4 14 Sly Cea ee Ly, 30, 37 EE epee 8 16 tae geet 18 36, 40 Fl Ht ee 9 19 | Re Suse. 19 39, 41 A cess bs 10 20,22 An examination of the table shows that the active elements may be classified in different ways :— (1) Active elements are odd-numbered elements in a regular sequence of numbers, viz., 5, 7, 9, 11, 13, 15. (2) The atomic masses of the active elements are given by 4n+a where nis a whole number ; a=3 forall the elements except nitrogen, for which it is 2. (3) With the exception of boron, which has two isotopes 10, 11), the active elements are all pure elements. P We have seen that no evidence has been obtained that the preceding element lithium (3), and the succeeding elements, chlorine (17) and potassium (19), show any trace of activity under a-ray bombardment, although they are odd-numbered elements and the masses of their isotopes are given by 4n+a. Magnesium and silicon, which are even-numbered, but which contain isotopes of mass 4n+1 or 4n+2, show no sign of activity. There thus appears to be no obvious general relation which differentiates active from inactive elements. The activity starts sharply with boron and ends abruptly with phosphorus. It is a very unexpected observation that neither lithium nor chlorine shows any certain evidence of activity in the emission of either long-range or short-range particles. It is of interest to consider whether any deduction can be made as to the structure of these nuclei in the light of these experimental facts. In our previous paper it was pointed out that the H nuclei A430 ~=6Sir E. Rutherford and Dr. J. Chadwick on the | liberated from the active elements probably existed as satellites circulating in orbits round the main nucleus, In the case of an effective collision of an a@ particle with such a nucleus, part of the momentum of the « particle is com- municated to the central nucleus, but the satellite is sufficiently distant from the latter to acquire enough momentum and energy to escape from the system. It was shown that such a point of view offers a general explana- tion of the variation of the velocity of the expelled H nuclei with the speed of the « particle and also of the escape of the H nuclei in all directions, The chance of ejecting an H satellite at high speed from a nucleus is much smaller (for nitrogen, for example, about 1/20) than the chance of setting a free H nucleus in correspondingly rapid motion, It appears therefore that the release of the satellite only takes place under certain restricted conditions of the collision of the « particle with the nucleus. If the H satellites were present in lithium and chlorine and were very lightly bound to the nucleus, it is to be anticipated that the number released by the « rays would be of the same order of magnitude as if the H nuclei were free. As this is found not to be the case, we may conclude that neither lithium nor chlorine has any lightly bound satellites in its nuclear structure. The complete absence of long-range particles from these elements shows that the H satellites, if they are present at all, are strongly bound to the main nucleus. If, for example, the satellite revolves very close to the nucleus, the « particle may only be able to give such a small part of its momentum to the satellite that it is unable to release it from the system. It does not, however, seem likely that the forces binding a satellite would vary greatly in passing from phosphorus to chlorine. It seems more probable that the general structure of the chlorine nucleus differs in some marked way from that of the group of active elements. The H nuclei may perhaps be definitely incor- porated into the main nuclear system, so that the a particle has no chance of concentrating its energy upon a single unit of the nuclear structure. In a similar way it seems probable that lithium must differ widely in structure from the suc- ceeding element boron. ‘The facts brought to light in these experiments indicate that the nuclei even of light elements are very complex systems and illustrate how difficult it will be to find any simple and general rule to account for the variation in structure of successive elements. It has been pointed out that, with the exception of the first Disintegration of Hlements by a Particles, 431 element boron, all the active elements are.“ pure” elements, 2. €., have no isotopes. This may be of some significance in differentiating between the structure of active and inactive elements, ‘The absence of isotopes indicates that, as regards mass, there is only a narrow range of stability of the nucleus for a given nuclear charge; the addition or subtraction of an equal number of H nuclei and electrons leads presumably to an instability of the nuclear system. In the case of lithium and of chlorine, which form isotopes, the forces binding the nuclei together may consequently be very different from those in the case of the pure active elements. If there is any significance in this point of view, it would indicate that H_ satellites are only present in pure odd- numbered elements; but, as we have seen, boron is an exception to this rule. In comparing the phenomena shown by the six active elements, it seems at once clear that nitrogen occupies an exceptional position in the group. Not only is the range of the expelled H nuclei the smallest of all the group, but the ratio of the ranges in the two directions is markedly different from those shown by the other elements. It is natural to connect this anomalous behaviour with the fact that the mass of the nitrogen nucleus is given by 4n+ 2, while the rest of the group are of the class 4n+3. The slower speed of ejection of the particles from nitrogen at first sight suggests that the H satellite is more hghtly bound than in the case of the other elements. This suggestion is, however, not borne out by calculation of the distribution of momentum among the three bodies involved in the collision, viz., the « particle, the H satellite, and the residual nucleus. In our previous paper, we showed that the distribution of momentum could be calculated on certain assumptions from the observations of the ranges of the expelled nuclei in the forward and reverse directions of the a particle. It was supposed that the law of conservation of momentum holds, and that the sum of the energies of the H particle and the residual nucleus was the same whether the H particle was liberated in the forward or backward direction. It follows from these assumptions that the relative velocity of the H nucleus and the residual nucleus is the same in thetwo cases, The results of this calculation for the group of active elements are collected in the following table (Table IT.). 432 The Disintegration of Elements by « Particles. TABLE IT. Distribution of Momentum. H particle. Residual Nucleus. a Gainin Hlement. forward. Backward. Forward. Backward. particle. Energy- Borow peepee... 202V —-175V —d4V. S23V 252V 427% Nitrogen ...... 78 V —1:32V Pa Ve 4b Vi ‘78 V -13 % Fluorine ...... 210V* -189V —J0V 389V 200V 35% NOGIUM As... ..5: 202V —172V 1A 2V 7) OG, ‘56 V 6 % Aluminium... 234V —211V “8V + 5:23.V. 788 V wet Phosphorus... 210V —189V 114V 513V -76V 15% The momenta are expressed in terms of the initial velocity V of the « particle. ‘ne initial momentum of the a particle, and consequently the sum of the momenta of the three bodies after collision, is therefore 4V. Momenta in the direction of the incident « particle are taken as positive, momenta in the opposite direction as negative. The percentage energy, gained from the nucleus as a result of the disintegration, is given in the last column, in terms of the initial energy of the « particle. It wiil be seen that in the case of nitrogen a considerable part of the momentum of the @ particle is communicated to the main nucleus, a much greater part than in the cases of the adjacent elements boron and fluorine. This indicates that the H satellite of nitrogen is in relatively close proximity to the main nucleus. It will also be noted that while for the other elements there is a gain of energy from the disruption varying from 6 per cent. for sodium to 42 per cent. for. boron and aluminium, for nitrogen there is a loss of energy of 13 per cent. It is apparent from the above table that the distribution of momentum among the three bodies varies considerably for the different elements, but, in the absence of any definite evidence of the validity of the theory on which the calcula- tions are based, it seems inadvisable to discuss these differences in any detail at the present stage. Cavendish Laboratory, June 20, 1922. XLII. The Distribution of Hlectrons around the Nucleus in the Sodium and Chlorine Atoms. By W. LAwRENcE Braga, M.A., F.R.S., Langworthy Professor of Physics, The University of Manchester; R. W. James, J/.A., Senior Lecturer in Physics, The University of Manchester ; and C. H. Bosanquet, Jf.A., Balliol College, Oxford*. 1. J N two recent papers f in the Philosophical Magazine the authors have published the results of measure- ments made on the intensity of reflexion of X-rays by rock-salt. The mathematical formula for the intensity of reflexion, as calculated by Darwin {, involves as one of its factors the amount of radiant energy scattered in various directions by a single atom when X-rays of given amplitude fall upon it. The other factors in the formula can be evaluated. By measnring the intensity of reflexion experi- mentally we can therefore obtain an absolute measurement of the amplitude of the wave, scattered by a single atom, in terms of the amplitude of the incident radiation. This measurement is of considerable interest, because it may throw some light on the distribution of the electrons around the nucleus of the atom. We regard the wave scat- tered by the atom, as a whole, as the resultant of a number of waves, each scattered independently by the electrons in the atom. A formula first evaluated by J. J. Thomson is used in order to calculate the amplitude of the wave scattered by a single electron. If an incident beam of plane polarized X-rays consists of waves of amplitude A, then the amplitude A' at a distance R from the electron in a plane containing the direction of the incident radiation, and at right angles to the electric displacement, is given by Pel Mahe weet AY > Rone’ ° . ° e ° . (1) Here e and m are the charge and mass of the electron in electromagnetic units, and ¢ is the velocity of light. What we measure experimentally is the resultant ampli- tude of the wave-train scattered in various directions by a number Z of electrons in the atom. If all the electrons were * Communicated by the Authors. + Phil. Mag. vol. xli. March 1921; vol. xlii. July 1921. ft C. G. Darwin, Phil. Mag. vol. xxvii. pp. 315-675 (Feb. and April 1914). Plul. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2F 434 Prof. W. L. Bragg and Messrs. James and Bosanquet : concentrated in a region whose dimensions were small com- pared with the wave- “length of the rays, then the resultant 2 amplitude would be equal to a. since the scattered wavelets would be in phase with each other in all directions. It is found experimentally that the measured amplitude tends to a value which is in agreement with the formula | at small angles of scattering, but that at greater angles it falls to a very much smaller value. ‘This is to be accounted for by the action of interference between the waves scattered by the electrons in an atom, which are distributed throughout a region whose dimensions are large compared with the X-ray wave-length. It is an easy matter to calculate the average Be His scattered in any direction by a given distribution of electrons around the nucleus. Here we are attempting to solve the reverse of this problem. The experimental results tell the amplitude of the wave scattered by the sodium and chlorine atoms through angles between 10° and 60°. We wish to use these results in order to get some idea of the manner in which the electrons are distributed. 2. In addition to Darwin’s original mathematical treat- ment, the question of the effect on X-ray reflexion of the distribution of electrons around the atom has been dealt with by W. H. Bragg *, A. H. Compton f, and P. Debye aG yey “Scherrer f. W.H. Bragg considered the inter one wien of the diminu- tion in the intensities of reflexion by a crystal as the glancing angle is increased, due allowance being made for the arrangement of the atoms. He concluded that “an ample explanation of the rapid diminution of intensities is to be found in the highly probable hypothesis that the scattering power of the atom is not localized at one central point in each, but is distributed through the volume of the atom.” He did not regard the experimental data then available as sufficient to justify making an estimate of the distribution of the electrons. These data indicated that the intensity of reflexion fell off roughly as a (0 being the glancing angle), and he showed that a density of distribution of the electrons could be postulated which * W. H. Bragg, Phil. Trans. Roy. Soc. Series A, vol. cexv. pp. 253-274, July 1915. + A. H. Compton, Phys. Rev. vol. ix. no. 1, Jan. 1917. { P. Debye and P. Scherrer, Phys, Zeit. pp. 474-4838, July 1918. Distribution of Electrons in Na and Cl Atoms. 435 accounted for this law, just as an illustration of the appli- cation of the principle involved in considering spatial distribution. A. H. Compton used the experimental results obtained by W. H. Bragg in order to calculate the electron distribution. W. H. Bragg showed that the intensity of reflexion is a function of the angle of reflexion alone, when allowance has been made for the arrangement of the atoms in the crystal, and he determined the relative intensity of reflexion by a number of planes in rock-salt and calcite. Compton cal- culated from these values the relative amplitudes of the waves scattered by the atoms in different directions, by means of the reflexion formula of Darwin, and proceeded to test various arrangements of electrons in order to find one which gave a scattering curve agreeing with that found experimentally. He supposed that the electrons were rotating in rings, governed by Bohr quantum relationships In sodium, for example, he placed four electrons on an inner ring, six on the next ring, and a single valency electron on an outer ring. In chlorine the rings contained four, six, and seven electrons respectively. Compton found that these atomic models gave a fair agreement with W. H. Bragg’s results. Debye and Scherrer came to the same conclusion as to the significance of intensities as regards electron distribution which was implied in W. H. Bragg’s work and stated more Fully by Compton. They considered two interesting cases. The first was that of the lithium fluoride crystal. They compared the intensity of reflexion by planes where the fluorine and lithium atoms reflected waves in phase with each other, with that by planes where these atoms acted in opposition to each other. The relative amplitudes at any F+i ov where F and Liare the amplitudes contributed by the fluorine and lithium atoms respectively. Their figures indicated angle for such planes may be expressed by the ratio that the limiting values of su at zero angle of scattering is 1*5, signifying that a valency electron has passed from the lithium to the ae atom (F925 = 15) : Their intensities of reflexion ey) measured by the darkening of a photographic plate in the powder method of analysis which these authors initiated. In view of the 22 436 Prof. W. L. Bragg and Messrs. James and Bosanquet : difficulties of estimating intensities in this way, of the few points which they obtained on the curve for the oa ratio, of the difficulties in interpreting intensities which we have discussed in our papers, and of the large extrapolation which they had to make in order to get the limiting value if F+hi i eat proving that the transference of the valency electron has taken place. The fact of the transference is supported by much indirect evidence, and their conclusion is probably correct. Debye and Scherrer also compared the intensities reflected by various planes of the diamond, and concluded that the electrons in the carbon,atoms were contained within a sphere of diameter 0°43 A, assuming a uniform distribution throughout this sphere. | In all the above cases, the results were obtained by com- paring the relative intensities of reflexion by various faces. The results which we have obtained, and which will be used to calculate the distribution of electrons in sodium and chlorine, are, on the other hand, absolute determinations. The intensity of reflexion was compared in each case with the strength of the primary beam of X-rays, so that the absolute efficiency of the atom as a scattering agent could be deduced. _ In a paper on “The Reflection Coefficient of Monochro- matic X Rays from Rock Salt and Calcite” *, Compton made comparisons of the incident and reflected beam, for the first order reflexion from cleavage faces of these'erystals. He obtained results for rock-salt which were rather less than those which we afterwards obtained for a ground face, but he noted that the effect was increased by grinding the face. In our notation the results were we feel that their results cannot be regarded as Compton - = 00044 + :00002 ; NaCl‘(100). Ko. i B.J.and B. “7 -= 00055 As Compton surmised, and as we have found experimentally, this figure for the efficiency of reflexion has to be modified considerably to allow for the extinction factor. The difference * A, H. Compton, Phys. Rev. vol. x. p. 95, July 1917. Distribution of Electrons in Na and Cl Atoms. 437 between his results and ours is accounted for by the extinction or increased absorption of the rays at the reflecting angle. Compton pointed out that the reflexion factor was of the order to be expected from Darwin’s formula, but did not use the value he obtained to solve the electron-distribution problem. 3. For the sake of convenience of reference, the formula which forms the basis of all the calculations is quoted below. Let the intensity I) of a beam of homogeneous X-rays, at a given point, be defined as the total energy of radiation falling per second on an area of one square centimetre at right angles to the direction of the beam. Ifa crystal element of volume dV, supposed to be so small that absorption of the rays by the crystal is inappreciable, be placed so that it is bathed by the X-rays, and if it is turned with angular velocity w through the angle at which some plane in it reflects the X-rays about an axis parallel to that plane, the theoretical expression for the total quantity of energy of radiation reflected states that a0: IN? Xx? 4 1+4cos? 20 san Oe oe ae Og « (2) resin 20 mc 2 =a V. In this expression N= Number of diffracting units per unit volume*, X% = Wave-length of X-rays. ? = Glancing angle at which reflexion takes place. e = Electronic charge. m= Electronic mass. c = Velocity of light. The factor e—Bsi""? (the Debye factor) represents the effect of the thermal agitation of the atoms in reducing the intensity of reflexion. The factor F depends on the number and arrangement of the electrons in the diffracting unit. At @=0 it would have a maximum value equal to the total number of electrons in the unit, and it falls off owing to interference as @ increases. The experimental observations have as their object the determination of @ in absolute units. In practice we cannot use a single perfect crystal so small that absorption is * No account is taken here of the “structure factor.” The diffracting units are supposed to be spherically symmetrical as regards their diffraction effects. 438 Prof. W. L. Bragg and Messrs. James and Bosanquet : . inappreciable. We use a large crystal consisting of num- bers of such homogeneous units and deduce, froin its reflecting power, the reflecting power Q per unit volume of the units of which it is composed. The assumptions made in doing this are by no means free from objections, and will be discussed later in this paper. Taking this to be justifiable, however, our experimental results yield the value of Q for rock-salt over a wide range of angles, and from them the values of Fo, and Fy, follow directly. These values are shown in fig. 1. Fig. 1. Values of F (4) Fo) corrected for Debye factor. (c) Fy, corrected for Debye factor. (6) Fy uncorrected __,, 5 (d) Fy, uncorrected __,, ” 4. We must now consider more closely the significance of the factor F. The most simple case is that of a crystal con- taining atoms of one kind only. Parallel to any face of the Distribution of Electrons in Na and Cl Atoms. 439 crystal we can suppose the atoms all to lie in a series of planes, successive planes being separated by a distance d. We get the nth order spectrum formed at a glancing angle by the reflexion from such a set of planes if 9d sin @=n\X. This spectrum represents the radiation diffracted by the atoms in a direction making an angle 20 with the incident beam, and it is formed because in this particular direction the radiation scattered by any pair of atoms lying in suc- cessive planes differs in phase by 2n7. Thus the amplitude of the beam scattered in this direction is the swm of the amplitudes scattered by all the neighbouring atoms taking part in the reflexion. Let us consider the contribution to the reflected beam of a group of atoms lying in a reflecting plane. To obtain the amplitudes of the reflected wave, we sum up the amplitudes contributed by the electrons in ail the atoms, taking due account of the fact that the electrons do not in general lie exactly in the reflecting plane and so contribute waves which are not in phase with the resultant reflected wave. By symmetry, the phase of the resultant wave will be the same as that reflected by electrons lying exactly in the geometrical plane passing through the mean positions of all these atomic centres. The phase of the wave scattered in a direction @ by an electron at a distance x from the plane differs from that of the resultant wave by an amount Agr 4% sin @. We will suppose that there is in every atom an electron which is at a distance a from the centre, and that all direc- tions of the radius joining the electron to the atomic centre are equally likely to occur in the crystal. In finding the effect of these electrons for all atoms (M in number) of the group, we may take it as equivalent to that of M electrons distributed equally over a sphere of radius a. It can easily be shown that, if z is the distance of an electron from the plane, all values of 2 between +a and —a are equally likely for both cases. Such a shell scatters a wave which is less than that scattered by M electrons in the plane in the ratio sin | p , where o¢= =) sin 0. 440 Prof. W. L. Bragg and Messrs. James and Bosanquet : The average contribution of the electron in each atom to the fF factor is therefore gut and not unity as it would be if the electron were at the centre of the atom. If there are n electrons at a distance a from the centre of the atom, their contribution to the F factor would be oe 5 5 gt ls Any arrangement of n_electrons at a distance a from the centre of the atom, provided that all orientations of the arrangement were equally probable, would make the same contribution to the F factor. For example, eight electrons arranged in a ring about the nucleus would give the same value for F as eight electrons arranged at the corners of a cube, or eight electrons rotating in orbits lying on a sphere of radins a. This illustrates the limitations of our analysis, which cannot distinguish between these cases. We can only expect to get information from our experimental results as to the average distance of the electrons from the atomic centre, and this for the average atom. Suppose now that any atom contains a electrons at a distance 7, from the nucleus, 6} at a distance 72, ¢ at a dis- tance 73... ata distance 7,, then the value of F tor the average atom would be given by pee phe foe +.. oe a (4) Thus, given the distribution of the electrons on a series of shells or rings, we can calculate the value of F for any value of 8. The problem we have to solve here, however, is the converse of this. We have measured the value of F fora series of values of 6, and wish to determine from the results the distribution of the electrons. We have seen above that there is no unique solution of this problem, but we can get some idea of the type of distribution which will fit the experimental curves. In order to do this, we suppose the electrons to lie on a series of sheils, of definite radii 7), 7), .... and determine the number of electrons a, 0, ¢ on the various shells which will give values of F corresponding to those observed experimentally. Suppose, for example, we take six shells uniformly spaced over a distance somewhat greater than Distribution of Electrons in Na and Cl Atoms. 441 the atomic radius is expected to be. For any given value of 6 we have | mee sin d; 4 pein ba Zan sin 3 b )3in py ie sin bs $1 $2 $3 di bs sin on ae Syl aes (5) 8 We chose from the experimental curve six values of 0 evenly spaced over the range of values at our disposal, and for each of these values read from the curve the value of F. Since definite radii have been assumed for the shells, the nae ete., can be calculated for each value of @. Hence, for sone value of 0, we have an equation involving numerical coefficients and the quantities a, b, ¢, d, e, 7, so that if six such equations are formed we may calculate these quantities. If Z is the total number of electrons in the atom we have BOGE OG ad Pet ct. 2. fe an 1 (6) and this will be taken as one of our equations (corresponding to @=0). In calculating the results for sodium and chlorine we have assumed the atom to be ionized, and bave taken Zo = 18 and _Zy, = 10. It wil be evident that this method of solution is somewhat arbitrary, and that the results we get will depend on the particular radii assumed for the shells. By assuming various radii for the shells, however, and solving the simultaneous equations for the number of electrons on each, we find that the solutions agree in the number of electrons assigned to various regions of the atom. As a test of the method of analysis, a model atom was taken which was supposed to have electrons arranged as follows :— 2 ona shell 0:05 A radius. DP Men rs Oren) Se ae ORL ON hates The F curve for this model was calculated. Then the simul- taneous equations for the electron distribution were solved, just as if this curve had been one found experimentally. This was done for two arbitrarily chosen sets of radii, taken out to well beyond the shell at 0°70 ye EE a 442 Prof. W. L. Bragg and Messrs. James and Bosanquet : The comparison between the two analyses (dotted curves) and the atom model we started with (continuous curve) is shown in fig. 2. The abscissee represent the radii of the shells in A, the ordinates the total number of electrons inside a shell of that radius. When the limits of the atomic structure are reached, the curve becomes horizontal at the value 10, corresponding to the ten electrons. The analyses not only indicate with considerable accuracy the way in Fig, 2. ) 0-5 1-5 2:0 25 Radius #4 of sptere, measured in Angstrom units. Number of electrons inside a sphere of radius 7. which the electron-content grows as we pass to spheres of larger radii, but also tell definitely the outer boundary of the atomic structure. Both give a number of electrons very nearly equal to zero in the ahote outside 0°70 A. 5. The F curves for sodium and chlorine can be solved in the same manner. We have expressed our results in two ways. First, we have supposed the electrons to be grouped on shells. The numbers of electrons on each shell, and the radii of the shells, have been so adjusted as to give the best possible fit to the experimental curves. In the case of sodium it is found that a fit can be obtained with two shells, and in the case of chlorine with three shells. The Distribution of Electrons in Na and Cl Atoms. 443 numbers of electrons on each shell, and the radii of the shells, are as follows :-— Sodium. 7 electrons on a shell of radius 0°29 A. 3 be i SO Or,s Chlorine. 10 electrons on a shell of radius 0°25 A. 5 mt ‘3 ) UTGG5 5 3 uF a Be ci | nee Secondly, we have solved the simultaneous equations for the distribution in shells with several sets of radii, and drawn a smooth curve through the points so obtained in such a way as to represent the density of distribution of the electrons as a continuous function of the distance from Fig. 3. 1 (£lectron density per Angstrom unit). Oistance From centre of atom in Angstrom waits, the atomic centre. The density P is so defined thatZPdr is the number of electrons whose distance from the centre lies between r and r+dr. The curves which we obtain for sodium and for chlorine are shown in figs. 3 and 4. The 444 Prof. W.L. Bragg and Messrs. James and Bosanquet: total number of electrons in the atom is represented by the area included between the curves and the axis. Fig. 4, ee. a P (Electron density per Angstrom unre). . e fe) oy ° Distance from centre of atom in Angstrom units. The following table shows the agreement between the F curves found experimentally and those calculated from the electron distributions :— TasLE I.—Sodium. Sin 6, 0-1. 0-2, 0:3. 0-4, 0-5. (Observed: on... 832 B40. 887) 202) aman F shells {234 \ 8°56 5:59 3:33 219 0-98 (Smooth Ourve .. 837 540 3290 1-91 1-00 TaBLE II.—Chlorine. Sin 9. 0-1. 0:2. 0:3. 0-4. 0:5. / Observed Wag se ah: 12°72 7°85 5:79 4°40 3:16 fe) | (025A) F 4 Shells ee i 13°53 (ule 5:90 4°61 2°69 | | 1:46 | Smooth Curve ... 12°70 7°80 B55) 410 3°20 6. We have also made an approximate calculation ef the F curve to be expected from an atom of the type pro- posed by Bohr*. In the ionized sodium atom containing * Nature, cvil. p. 104 (1921). Distribution of Electrons in Na and Cl Atoms. 445 10 electrons, two are supposed to describe circular one- quantum orbits about the nucleus, while, of the remaining eight, four describe two-quantum circular orbits and four two-quantum elliptical orbits. We have calculated the size of these orbits from the quantum relationship and the charges ; this can only be done very approximately, owing to the impossibility of allowing for the interaction of the electrons. We take the following numbers :— Radius of 1 quantum ring ...... 0:05 A. $5 2 AA aN aes te 0°34 ,, Semi-major axis of ellipses * ... 0°42 ,, To get a rough idea of the diffracting power of such an atom, we suppose, first, that the orientation of the orbits is random so that the average atom has a spherical symmetry, and also that the periods of the electrons in their orbits are so large compared with the period of the X-rays that we need not consider the effect of their movements. The calculation of the effect of the circular orbits offers no difficulties. To allow for the effect of the ellipses, the following method was used. The elliptical orbit was divided into four segments, through each of which the electron would travel in equal times. It was then assumed that, on the average, one of the four electrons describing ellipses would be in the middle of one of these segments. This gives four different values of the radius vector, corresponding in the average atom to four spherical shells of these radii. We thus calculate the value of F for an atom having 2 electrons on a shell of radius 0°05 A.U. relate £ Aen egeei arc. 1 i Maer Op oa. fe Piling ; a Se gens Ou be fbi, i aac OSTEO 2 Re ! m Owe ky * The elliptical two-quantum orbit of a single electron about the: sodium nucleus would have a semi-major axis equal to the radius of the two-quantum circle. We have used the larger value 0°42 to make- some allowance for the fact that part of the orbit lies outside the inner electrons, so that the effective nuclear charge is reduced. 446 Prof. W. L. Bragg and Messrs. James and Bosanquet: This gives the following figure for Fy, :— Sin @. 0-1. 0:2. 0:3. 04. 0°5. F calculated ............ 8:73 5-04 3-76 2°58 1:80 IPO HSELVEC ) iii). icieosctuease 8°32 5:40 3'37 2°02 0:76 The agreement, of course, is not perfect, but one must remember that no attempt has been made to adjust the size of the orbits to fit the curve. The method of calculation too is very rough, although it must give results of the right order. ‘The point to be noticed is that the curve is quite of the right type, and there is no doubt that an average distri- bution of electrons of the nature given by such an atom model could be made to fit the observed value of F quite satisfactorily. 7. The points which appear to us to be most doubtful in the above analysis of our results are the following :— (a) We have assumed that each electron scatters inde- pendently, and that the amount of scattered radiation is that calculated for a free electron in space according to the classical electromagnetic theory. It is known that for very short waves this cannot be so, since the absorption of y rays by matter is much smaller than scattering would account for, if it took place according to this law. On the other hand, the evidence points towards the truth of the classical formula, in the region of wave-lengths we have used (0°615 A). (6) We have used certain formule (given in our previous papers, to which reference has been | made) in order to calculate the quantity we have called @ in equation (2) from the observed intensity of reflexion of a large crystal. Darwin * has recently discussed the validity of these formule. The difficulty lies entirely in the allowance which has to be made for “extinction” in the crystal. X-rays passing through at the angle for reflexion suffer an increased absorption owing to loss of energy by reflexion. Darwin has shown that this extinction._is of two kinds, which he has called primary and secondary. If the crys- talline mass is composed of a number of nearly- parallel homogeneous crystals, each so small that absorption in it is inappreciable even at the reflecting angle, then secondary extinction alone takes place. At the reflecting angle the * Phil. Mag. vol. xliii. p. 800; May 1922. Distribution of Electrons in Na and Cl Atoms. 447 X-rays suifer an increased absorption, because a certain fraction of the particles are so set as to reflect them and divert their energy. We made allowance for this type of extinction in our work, and Darwin concludes that our method of allowance, while not rigorously accurate mathe- matically, was sufficiently so for practical purposes. Primary extinction arises in another way. The homo- geneous crystals may be so large that, when set at the reflecting angle, extinction in each crystal element shelters the lower laye ers of that element from the X-rays. Darwin has calculated that this will take place to an appreciable extent for the (100) reflexion if the homogeneous element is more than a few thousand planes in depth. A large homogeneous element such as this does not produce an effect proportional to its volume, since its lower layers are ineffective, and a crystal composed of such elements would give too weak a reflexion. Our method of allowing for extinction will not obviate this effect. We cannot be sure, therefore, that we have obtained a true measure of Q for the strong reflexions. The F curve may be too low at small angles. It is just here that its form is of the highest importance in making deductions as to atomic structure. Until this important question of the size of the homogeneous elements has been settled, we must regard our results as provisional. (c) The allowance for the thermal agitation of the atom {the Debye factor) is only approximate; it depends on a few measurements made by W. H. Bragg in 1914. In order to see how much error is caused by our lack of knowledge of the Debye factor, we have calculated the electron distribution without making any allowance for it. The result may appear at first rather surprising ; the electron distribution so calcu- lated is almost indistinguishable from that which we found before, when allowance for the Debye factor had been made. This is so, although the factor is very appreciable for the higher orders of spectra, reducing them at ordinary tem- peratures to less than half the theoretical value at absolute zero. The difference which the factor makes can best be shown by comparing the radii of the shells which give the best fit with (1) the F curve deduced directly from the expe- rimental results, (2) the F curve to which the Debye factor has been applied. 448 Prof. W. L. Bragg and Messrs. James and Bosanquet : (1). | (2). Radius Radius (without allowance (with allowance for thermal for thermal agitation). agitation). Sodium.—Seven electrons . ... Oa 0°29 Three electrons...... 0-79 0-76 Chlorine.—Ten electrons ...... 0°28 0°25 Seven electrons...... 0’81 0:86 Three electrons.. ... 1:46 1-46 A little consideration shows the reason for this. The form of the F curve at large angles is almost entirely decided by the arrangement of the electrons near the centre of the atom. A slight expansion of the grouping in this region causes a large falling off in the intensity of reflexion. This is shown in the analysis by the slight increase (0:02 to 0:03 A) in the radius of the shell which gives the best fit to the uncorrected curve. ‘!he effect of ihe thermal agitation is to make the electron distribution appear more widely diffused ; however, the average displacement of the atom from the reflecting plane owing to its thermal movements is only two or three hundredths of an Angstrém unit at ordinary temperatures, and so we get very hiiie alteration in our estimate of the electron distribution. The uncertainty as to the Debye factor, therefore, does not introduce any appreciable error in our analy sis of electron distribution. 8. It is interesting to see whether any avaclenee can be obtained as to whether a valency electron has been trans- ferred from one atom to the other or not. This may be put in another way: can we tell from the form of the F curves in fig. 1 whether their maxima are at 10 and 18 or at 11 and 17 respectively? It appears impossible to do this ; and, when we come to consider the problem more closely, it seems that crystal analysis must be pushed to a far greater degree of refinement before it can settle the point. If all the electrons were grouped close to the atomic centres, and if the transference of an electron meant that one electron passed from the Na group to the Cl group, then a solution along the lines of that attempted by Debye and Scherrer for LiF might be possible. The electron distributions we find extend, on the other hand, right through the volume of the orystal. The distance between Na and Cl centres is 253 iy and we find electron distributions 1 A from the centre in sodium and 1°8 A from the centre in chlorine. lf the ——- Distribution of Electrons in Na and Cl Atoms. 449 valency electron is transferred from the outer region of one atom to that of the other, it will still be in the region between the two atoms for the greater part of the time, since each atom touches six neighbours, and the difference in the diffraction effects will be exceedingly small. It is for this reason that we think Debye and Scherrer’s results for LiF, which were not absolute measurements such as the above, were not adequate to decide whether the transference of a valency electron has taken place. We have assumed that the atoms are ionized in calculating -vur distribution curves. If, on the other hand, we had assigned 11 electrons to sodium and 17 to chlorine, we should have obtained curves of much the same shape but with an additional electron in the outermost shells of sodium and one less in those of chlorine. 9. Summary.—We have attempted to analyse the distri- bution of electrons in the atoms of sodium and chlorine by means of our experiments on the diffraction of X-rays by these atoms. The results of the analysis are shown in figs. 3 and 4. The principal source of error in our conclusions appears to be our ignorance as to the part played by “extinction” in affecting the intensity of X-ray spectra. The distributions of the electrons are deduced from the F curves (fig. 1). The most important parts of these curves are the initial regions at small angles, for errors made in absolute values in this region alter very considerably the deductions as to electron distribution. The exact form of the curve at large angles is of much less interest. Now, it is in this initial region, corresponding to strong reflexions such as (100), (110), (222), that extinction is so uncertain a factor. Until the question of extinction is satisfactorily dealt with, the results cannot be regarded as soundly established.. If our results are even approximately correct, they prove an important point. There cannot be, either a sodium or chlorine, an outer “shell” containing a group of eight electrons, or eight electrons describing orbits lying on an outer sphere. Such an arrangement would give a diffraction eurve which could not be reconciled with the experimental results. Hight electrons revolving in circular orbits of the same radius would give the same diffraction curve as eight electrons on a spherical shell, and are equally inadmissible. On the other hand, it does seem possible that a combination of circular and elliptical orbits will give F curves agreeing with the observations. Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2G [ 450 ] XLIV. On the Partition of Energy. By C..G. Darwin, I.A., F.RS., Fellow and Lecturer in Christ’s College, Camb., and R. H. Fowier, M.A., Fellow and Lecturer in Trinity College, Camb.* §1. Introduction. A N important branch of atomic theory is the study of the way in -which energy is partitioned among an assembly of a large number of systems—molecules, Planck vibrators, etc. This study is based on the use of the principles of probability which show that one type of arrangement is much more common than any other. The most usual method is to obtain an expression for the probability of any state described statistically and then to make this probability a maximum. This always involves a use of Stirling’s approxi- mation for factorials, which in many cases is illegitimate at first sight, and though it is possible to justify it subsequently, this justification is quite troublesome. It is also usually required to find the relation of the partition to the temperature - rather than to the total energy of the assembly, and this is done by means of Boltzmann’s theorem relating entropy to prebability—a process entailing the same unjustified approxi- mations. The object of the present paper is to show that these calculations can all be much simplified by examining the average state of the assembly instead of its most probable state. The two are actually the same, but whereas the most probable state is only found by the use of Stirling’s formula, the average state can be found rigorously by the help of the multinomial theorem, together with a certain not very difficult theorem in the theory of the complex variable. . By this process it is possible to evaluate the average energy of any group in the assembly, and hence to deduce the relation of the partition to temperature, without the intermediary of entropy. The temperature here is measured on a special scale, which can be most simply related to the absolute scale by the use of the theorem of equipartition, and we shall also establish the same relationship directly by connecting it with the scale of a gas thermometer. Throughout the paper the analysis is presented with some attempt at rigour, but it will be found that apart from this rigour it is exceedingly easy to apply the method of calculation. Most of the results are not * Communicated by the Authors. ae On the Partition of Energy. 451 new; it is the point of view and the method which, we think, differ from previous treatments No discussion of the question of partition would be com- plete without consideration of its relation to thermodynamic principles. We shall leave this view of the subject to a future paper ; for the increased light thrown on the statistical nature of entropy raises many interesting points which could not be discussed here properly without making the present work run to inordinate length. § 2. Statistical Principles and Weight. Before proceeding to the problem it will be well to review, in general outline, the principles of the theory of the partition of energy, though we have nothing new to say in this connexion. We shall be concerned with collections of molecules, Planck vibrators, etec.—each individual unit will be called a system, and we shall call the whole collection an assembly. We shall be dealing mainly with assemblies com- posed of groups of systems, the individuals in each group being identical in nature. In order to make the problem definite it is necessary to assume that each system has. a definite assignable energy, and yet can interact with the others. This requires that the time of interactions, during which there will be energy which cannot be assigned to a definite single system, is negligibly small compared with the time during which each system describes its own motion, For such an assembly we are to calculate various average properties of its state, when it describes its natural motion according to whatever laws it may obey. There will at any rate be an energy integral, and we have therefore to calcu- late these averages subject to the condition of constant energy. To determine the basis on which these averages are to be calculated we are to apply the principles of proba- bility ; and the calculation of itself falls into two stages, the prior and the statistical. The prior stage aims at establishing what are the states which are to be taken. as of equal probability. In the statistical stage we have simply to enumerate the states specified in the prior stage, allow for the fact that the systems are macroscopically indistin- guishable, and evaluate the averages taken over these states. It is not here our purpose to enter into a full discussion of the fundamental questions that arise in connexion with the determination of what states ought to be taken as equally 2G 2 452 Messrs. C. G. Darwin and R. H. Fowler on probable. It will suffice to recall tnat for assemblies obeying the laws of classical mechanics the theorem of Liouville shows that the elements of equal probability may be taken to be equal elements of volume in Gibbs’ “ phase space.” It follows out of this, for example, in the case of an assembly of a number of identical systems—say simple free mole- cules—that the elements of equal probability can be simpli- fied down into 6-dimensional cells dq,dq.dq3dp,dp.dp3 of equal extension, where 4, q, g3 are the coordinates, and Pi, Px» pz the conjugated momenta, of a single molecule. We shall describe this by saying that the weght of every equal element dg, ... dp; is the same, and by a slight generalization, that the weights of unequal cells are pro- portional to their 6-dimensional extension. The word weight is here used in exactly the sense of the term a priori probability, as used by Bohr and others. But when we come to the quantum theory, mechanical principles cease to hold, and we require a new basis for assigning the equally probable elements. Such a basis is provided by Hhrenfest’s * Adiabatic Hypothesis and Bohr’s + Correspondence Principle. These show how the theorem of Lionville is to be extended, and allow us to assign a weight for each quantized state of a system. It is found that we must assign an equal weight to every permissible state in each quantized degree of freedom. At first sight this is a little surprising, for it would seem natural to suppose that a vibrator which could only take energy in large units would be less likely to have a unit than one which could take it in small; but this is to confuse the two stages of the problem. It is only by the supposition of equal weights that we can obtain consistency with classical mechanics by the Corre- spondence Principle. It is customary { in assigning a definite weight to every quantized state to give it the value h, so as to bring the result to the same dimensions as those of the element dq dp in the classical case. But there is considerable advantage in reversing this, and taking the quantized weights as unity and the weight of the element in the phase space as dqdp/h; for if this is done, the arguments about entropy are simplified by the absence of logarithms of dimensional quantities. We shall adopt this convention here, though in * Ehrenfest, Proc. Acad. Amst. xvi, p. 591; Phil. Mag. xxxiil. p. 500 (LOM arene: { Bohr, “The Quantum Theory of Line Spectra,” Dan. Acad. iv. p, 1 (1918). { Ehrenfest & Trkal, Proc. Amst. Acad. Se, xxiii. p. 162. See in par- ticular p. 165 and Additional Notes, No. 1. the Partition of Energy. 453 all our results it is immaterial—indeed, until such questions as dissociation are considered it makes no difference to adopt different conventions for different types of system. The convention has the advantage of shortening a good many formulee and freeing them from factors which are without effect on the final results. An exception to the above rule for assigning weights to quantized motions occurs in the case of degenerate systems, where there are two degrees of freedom possessing the same or commensurable frequencies. In this case there is only one quantum number, and the state of the system is partly arbitrary. Bohr* shows that the rational generalization is to assign to such a state a weight factor which can be evaluated by treating the system as the limit of a non- degenerate system, and quantizing it according to any pair of variables in which it is possible to do so. The number of the permissible states which possess the same total quantum number will give the weight of the state. A corresponding rule holds for systems degenerate in three or more degrees of freedom. The meaning of weight can perhaps be made clearer by considering its introduction the other way round—beginning with an assembly of simple quantized systems of various frequencies. Given the energy, there is a definite number of possible states, which are fully specified by the energy assigned to each system. We then make the hypothesis that it is right to assign an equal probability to each such state in the calculation of averages. Thisis now the fundamental postulate. The generalization to degenerate systems goes as before, by introducing weight factors. Finally, passing over to mechanical systems, such as free molecules, we are led by an appeal to the converse of the Correspondence Principle to attach weight dg, ...dp3/h? to each 6-dimen- sional cell which specifies completely the state of a single molecule. The second, statistical, half of the problem consists in enumerating the various complexions possible to the assembly. By a complexion we mean every arrangement of the assembly, in which we are supposed to be able to distinguish the in- dividuality of the separate systems. We count up the total number of complexions which conform to any specified statistical state of the assembly, and attach to each the appropriate weight factor. Thus the probability of this state is the ratio of the number of its weighted complexions to the * Bohr, loc. cit. p. 26. 454 Messrs. C. G. Darwin and R. H. Fowler on total number of all possible weighted complexions. This part of the problem depends on the nature of the particular assembly considered, and so must be treated separately in each case. We start in §3 with a problem which concerns not the partition of energy, but the distribution of molecules in a volume. It illustrates the method in its simplest aspect and has the advantage of being purely algebraic. Next, in $4, we take the distribution of energy among a set of similar Planck vibrators, which is again a purely algebraic process, and then proceed in § 5 to introduce the main theme of this paper by dealing with the partition of energy between two sets of Planck vibrators of different type. ‘This is most conveniently treated by using the complex variable, and in § 6 there is a discussion of the required theorem. The par- tition of itself introduces the temperature, and in §7 the special scale is compared with the absolute. In $$9, 10, 11 the partition law is generalized to more complicated types of system, such as the quantized rotations of molecules. In §$ 12, 13 the method is extended so as to deal with the free motion of monatomic molecules, intermixed with vibrators. The work leads to a rather neat method of establishing the Maxwell distribution law. § 3. The Distribution of Molecules in Space. The first example we shall take is not one of a partition of energy, but of the distribution of small molecules in a vessel. It illustrates in its simplest form the averaging process, and has the advantage of depending only on elementary algebra. | Let there be M molecules, and divide the vessel into m cells of equal or unequal volumes 1, v2...Um, which may each be as large or as small as we like. Then Oy hte hy Um SVs Tee By well-known arguments which we need not consider, it follows that any one molecule is as likely to be in any element of volume as in any other equal one. So by a slight ex- tension of the idea of weight we attach weights vj, v2, ... Um to the cells. To specify the statistical state we say that the first cells has a, molecules, the second a2, and soon. Then Ay+ Ag+ Ae +a, M. wie ey het eta te (3°2) By the theory of permutations the number of complexions. the Partition of Energy. 455 which conform to the specification is M! NG Ogee and each of these must be weighted with a factor V1 V9"? eee Vin ™. The total number of all the weighted complexions is 1 | : faye hs Qi Api... = (vy + (op) =a paste a by the multinominal theorem. This could have been deduced at once by working direct with probabilities v,/V instead of weights v,, but the argument has been given in detail to illustrate the method for more complicated cases. We next find the average value of a,. This is given by Ca— > .G, Di pe? wae |s ay ! Ag ! ware To sum this expression we only have to cancel a, with the first factor in a,! in the denominator, and then it is seen to be equal to Mov,(v, + 2 +...)"74, and so, as is implicit in our assumptions, a> = Mv,/V. e e ° ° e . (3°3) But we can now go further and find the range over which a, will be likely to fluctuate. ‘This is estimated by averaging the square of the difference of a, from its mean value. We shall throughout this paper describe such a mean square departure as the fluctuation of the corresponding quantity. Thus the fluctuation of a, is (ar—a,)?. Now (a,— dy)? =a,(a,—1) +a,—2a a,+a,", and averaging the separate terms by the multinominal theorem, we have ———=, . M(M—1)r Mv, 2M», Mv, Mov, 2 co Sar ae OTe ae aa, Mv, Uy _ = ay F => (1-7) =4. (1-H). De gaan This result represents the fluctuation however large or small 456 Messrs. C. G. Darwin and R. H. Fowler on v, may be. In all cases we have the result that (a,—q,)? is less than a,, and therefore that the average departure of. a, from a, is of order (a,)?. We can also interpret this fact by saying that departures of a, from a. which are much greater than (a,.)? will be relatively rare ; as M is large and (a,)? small compared with a,, this is precisely equivalent to saying that the possession of: the average value of a, is a normal property of the assembly in the sense used by Jeans * We have thus a simple and. complete proof that uniform density is a normal property of this assembly. §4. The Distribution of Energy among a Set of Planck Vibrators. Another case where the treatment can be almost entirely algebraic is that of the partition of energy among a set of Planck vibrators which all have the same frequency. Let e be the unit of energy so that every vibrator can have any multiple of e. As we saw in § 2, the weight attached to every state is to be taken as unity. Let there be M vibrators and let there be Pe of energy (P is an integer) to be partitioned among them. To specify a statistical state, let a, be the number of vibrators with no energy, a, with e, a, with 2e, etc. Then we have dod +aota,+:..= My. « 1 ee Oy + 2ao 4+ 3G3 4 wes = Ps . e e . (4°2) and anv set of a’s which satisfy these equations corresponds to a possible state of the assembly. By the principles of § 2 each of the complexions will have unit weight. Now count up the number of complexions corresponding to the speci- fication. By considering the various permutations of the vibrators, it is seen to be M! a)! a! Gs (4°3) We must next find © the total number of all possible complexions. Let Sy segobe summation over all possible values of the a’s which satisfy (4:1) and (4:2). Then Os M! oh Cig icy | te Consider the infinite series (l+e+74294 ...)™ * Jeans, ‘Dynamical Theory of Gases,’ passim. the Partition of Energy. 457 expanded by the multinominal theorem. The typical term is M! Ty big Wey lew ~24+2do+ 343... oe 5) where the a’s take any values consistent with (4:1). Then if we pick out the coefficient of 2’, we have the sum of all the expressions for whieh the a’s satisfy both (4:1) and (4:2). Observe that we may take the whole infinite series because the later terms are automatically excluded. , Now this will be the coefticient of z? in (1—z)~™, and so gE Ee (M—1)! P!’ which is the ordinary expression for the number of homo- geneous products as formerly used by Planck *. We next evaluate the average of a,; (4-4) eat ao! al ae ERS VA = Me tates where >, denotes summation over all values satisfying Ay +a;' +a’ +a;'... =M—l, a,’ +2a,/+ 3a,’ ... =P—r. The sum is thus pl +P—r—2)! ‘ (M—2)!(P—r)!’ and we have M+ P—r—9?)! P! Ar — M(M 1! : : Meri (Pant This is exact, and holds for all values of r ; now 7 can have any value up to P and the majority of the a,’s will be zero. The ordinary method of proof applies Stirling’s formula for a,! to these zero values. In the important case where both M and P are large, it will be only necessary to consider values of + which are small compared with P. Now, if 7? is small compared with P, P!/(P—r)! has the asymptotic value P”. Using relations of this type and also disregarding the difference between M and M—1, we have JA We eee 3 (M+ prt ne Oe ee (4°5) * M. Planck in the earlier editions of his book on Radiation. ie 458 Messrs. C. G. Darwin and R. H. Fowler on The same methods give the fluctuations of a, For Cas — Ay)? = Ay Ap — 1) + Up — (Ge), and a process similar to the above gives ———= M+ P—2r—8)! NG _y@ Pitas ble Se) (M—23)h(P— art The exact expression for the fluctuation can be at once put down. When M and P are taken large the leading term cuts out, and so it is necessary to carry the approximation to the next order. If we substitute | P!/(P—r)! ~Pr—4$r(r—1)P"7}, we find that (4G) = | a4 Sar epee Ore \ (4:6) The formula for a, can be put into a more familiar form by the substitution P=M/(e*—1), which gives p= Mera — 65%), i.) and leads to a corresponding but more complicated ex- pression for the fluctuation. Here, as we shall see later, « can be identified with the familiar ée/kT. Equation (4'6) establishes at once that the statistical state specified by (4°5) or (4°7) is a normal property of the assembly. $95. The Partition of Energy among two Sets of Planck Vibrators. After these preliminary examples we now apply our method to a problem which will bring out its distinctive character, that of the partition of energy in an assembly composed of different types of system. We shall consider first the simplest of such cases—an assembly consisting of a large number of Planck vibrators of two types A and B. The number of A’s. is M, and the energy unit of an A is eas before. There are now also N B’s with energy unit 7. To make exchanges of energy possible we have to suppose, say, that there are present a few gas molecules, but that the latter never possess any sensible amount of energy. (Later on in § 12 we develop. a method by which we shall be able to include any number of such molecules in our assembly.) We also require for the purposes of the proof to assume that e and 7 are com- mensurable, but it does not matter how large the numbers “may be which are required to express the ratio e/n in its the Partition of Energy. 4.59 lowest terms. To avoid introducing new symbols, we may suppose that the unit of energy is so chosen that e and » are themselves integers without a common factor. We have already introduced the idea of weight, and seen that we must assign the weight unity to every permissible state of a linear vibrator. To calculate the number of com- plexions of the assembly of any given sort, we have merely to calculate the number of ways in which the energy may be distributed among the vibrators, subject to the given statistical specification. A simple example will make the process clear. . | Let there be two A’s and two B’s; let n=2e, H=4e. Then the possible complexions are :— eae aab / | Anat ed ee | aab a'a'a'a aa’b bb Yoel A AY pad aa b b'b' aay Oe) / aa’a’a aalb aaa'a’ aa’ b/ bb! Here, for example, aaaa’ means that there is 3e of energy on the first of the A’s, e on the second, and none on the B’s. Hach of the fourteen complexions is, by definition, of equal weight, and is therefore to be reckoned as of equal probability in the calculation of averages. Observe how with the small amount of energy available a good deal more goes into the smaller than into the larger quanta; for the pair of A’s have on the average +e, as against +e for the pair of B’s. We pass to the general case. The statistical state of the assembly is specified by sets of numbers a,, b; where a, is the number of vibrators of type A which have energy re, and 6, the number of B’s with energy sn. All weights are unity and the number of complexions representing this statistical state is the number of indistinguishable ways (combinations) in which M vibrators can be divided into sets of a, a,... and at the same time N into sets b), b, .... As illustrated by the example, it is therefore given by the formula M! N! Ay! ay! ay! Sins bWO;! fila: (5:1) In (5:1) a, and 6, may have any zero or positive values consistent with the conditions Ors M, 3.b=N, &,rear+S.snbs=H, .. (52) where E is the total energy of the system—necessarily an 460 Messrs. C. G. Darwin and R, H. Fowler on integer in the units we employ. The total number C of all complexions is therefore M! N! y Ore? a Vata, Ni, ea Pe) where the summation &,,, is to be carried out over all positive or zero values of a, and 6, which satisfy (5:2). By using (5°1) and (5°3) we can at once obtain an expression for the average value, taken over all complexions, of any quantity in which we are interested. We have already studied a, in §4. The main interest centres in Ba, the average energy on the A’s. We have at once (S,0e0,)M! N! Ay! ay! ay! wat by! by! be! eee ; CE,= 2.5 (5:4) The following process leads to simple integrals to express the quantities C, CE, etc. Consider the infinite series (L428+2% + 0... ya expanded in powers of z by the multinominal theorem. The general term is M! =p EA, Gehl 6 ~ It follows that if we select from the expansion of (L-pett 204.) ML 4284 et a) the coefficient of z®, we shall obtain the sum of all possible terms such as (5'1) subject to the conditions (5:2), that is to say C. Similarly, if we form the expression {oo (+s habe jc) el aoa the general term in the first bracket must be (Syreas)M! yyy a, ! ay! eee c and by the same reasoning the coefficient of 2* in (5°6) must be CHa. : Expressions (5°5) and (5:6) are easily simplified— they are respectively | it (1 —2*)-M(1—27)-9, {oF (1 —2)x| (L—27)-%, the Partition of Energy. 461 The latter can also be written as { —Mz“log (1—<2*) \ (1—2§) -M(1—21)-N, If these expressions are now expanded in powers of < by the binomial theorem, they give asum of products of factorials which are, of course, the “‘ homogeneous product ” expressions used by Planck. It is possible to approximate to these by a legitimate use of Stirling’s theorem and to replace the sums ae : Ee by integrals without much difficulty. It would, indeed, have been possibig to start from these expressions, but we have not done so because in the general case to be discussed later that method would not be available. To make further progress’ by a method of general utility, we discard Stirling’s theorem and express these coefficients of 0. We see at once that we can suppose that H?a ranges effectively over all values from —o to +00 while all other terms, such as a,..., Ea3,... remain small. We then obtain for (6°1) on putting z=Se” the asymptotic expression 5 Lo(s)]® ( (FS) +i9E (8) + O(c?) + Oat) jet, For most purposes the first term in the expansion will suffice, but if the precise values of the fluctuations are re- quired, the second also is necessary. As it is in general rather complicated, we shall content ourselves here with pointing out its order of magnitude. On carrying through the necessary calculations we find ge (9, ¢, F} | onnseTa lL — smstgrig| «+ 6® The argument of F and @ is everywhere 3; the term {3, h, F} denotes a complicated expression of 3, and its first four derivatives, and F and its first two derivatives, but, is independent of H. If condition (v.) is dropped, we shall have 8 equal maxima arranged round the circle y, and, pro- vided F' has the same value at each of them, the integral will have a value equal to (6:2) multiplied by @. Now consider the probiem of § 5, to which our work applies immediately with tba 27 (dee) ee et) Nog LY ets ea We may suppose that E tends to infinity and also M and N the Partition of nergy. 465 in such a way that M/E and N/E are constant. This func- tion satisfies all the conditions of this section—the fact that it is in general many-valued is irrelevant, for we are only concerned with that particular branch which is real when z is real and 0,rea,'+%,snb,=H—re. Applying the multinomial theorem and reducing the the Partition of Energy. 467 expression to a complex integral, we have A) Ce amie (ley ri —2")" which, by virtue of the value of C from (5'71) and the argument of § 6, at once yields a,=Ms"(1—$°), =Me-"k2(1—e-ekT), | 2. . (83) which is the formula of § 4 over again, the presence of the B’s being immaterial. When we come to evaluate fluctuations the matter is a little more complicated, because the leading terms cut out, and so the second term of the asymptotic expansion will in general play a part. For example, consider the fluctuations of a,: | (4,—4,)°=4,(4,— 1) +4a,— (a,)*. By arguments exactly similar to those above, we have ‘ee 1 7 ae Bee Ca,(a,—1)=M(M—1)5— eg ee se and so by (6°2) ; a,(a,,— 1) =M(M—1)3""*(1—3*)’$14 O(1/E)}. Thus the fluctuation is a, —M3$*’*(1—3*)? + O(M?/E). This is sufficient to show that the possession of a, (8°3) is a normal property of the assembly. The complete calculation of the O-term is rather com- plicated; the result is given at the end of this section. But a great simplification arises if we suppose that there are many wore B’s than A’s, while E is so adjusted that $ the temperature is unchanged. In this case the term O(M?/E) becomes small and may be neglected. We shall describe this case by saying that A is in a bath of temperature 4S. Then, provided this is so, we have (a) =e ae VEy 2 |. (84) A much more important quantity is the fluctuation of E,. This is found by evaluating H,”. Now, just as CH, was given by operating with zd/dz on the first factor in 2 2 oa fd 468 Messrs. O- G. Darwin and R. H. Fowler on (eae jo le en tee CE,2 is easily seen to be given 2 by operating with (:5) in the same way. Thus 9 1 @ dz a \e €\— = CE) na) te 271 ve If we again suppose an infinite bath of we 3, we can omit the second term of the asymptotic expansion G 2) and obtain Be=(1-9°)* Gas) (i—sy™}, = (1— ae = BE, (1—9*)-™}, =(E,)?+3—5) and so the fluctuation is (E,—B,)?=E,?—(H,)?,4 ean = erage : (8°5) Ege 4 This is a result of which Einstein * made use in his work on fluctuations of radiation. It should be emphasized that these results are only accurate in a temperature bath, and not when the number of systems A is a finite fraction of the assembly. ; a In all cases (6:2) shows that the possession of Hy, is a normal property of the assembly. If we work out exactly the second terms in the asymptotic formuls of § 6 and apply them to the fluctuations of a, and Ei, we find 3 re — Hi, /M (a,—4,)° =a, [1-9 {1+ Me En y |. (8°6) (n= oe soe 94 lh . 6 gee Formula (4°6) above is a special case of (8°6). * A. Einstein, Phys. Zettschr. vol. x. p. 185 (1909). the Partition of Energy. 469 Finally it is of some interest to point out that we can obtain a formula for (H,—H,)** of general validity. We have in fact (Ba—Ha)*=1.3...(2s—1){(Ea—Ba)?}*, . (8°8) where (E,—E,)? is given by (8°7). We retain of course only the kighest order term *, which is thus O(E,)*. § 9. Generalization to any number of types of system, and to systems of any quantized character. It is clear that the present method of treating partitions is of a much more general character than has so far been exhibited. Consider an assembly composed of two types, A and B, of quantized systems more complicated than Planck vibrators. We suppose generally that the systems of type A, M in number, can take energies to the extents €9, €, €, ..., and these states have weight factors py, 1, Ps, ... in conformity with the discussion in § 2. Similarly, the B’s, N in number, can take energies , 7, M2, ... with weights qo, q1, Ya, --.- We have to suppose that it is possible to determine a basal unit of energy such that all the e’s and 7’s can be expressed as integers. Further, it simplifies the work if we suppose that there is no factor common to all of them. Proceeding exactly as before, we set down the weighted number of com- plexions which correspond to the specification that, of the A’s, a, have energy e,; of the B’s, b, have energy n,. This number is M! ts Us N! pate Elance a ee Cao ey Thies and the a’s and 0’s are able to take all values consistent with 2 ¢—=M, 2ba=Ne peat Sy b= Hh. ©. (92) Tengen ts Now form the functions ie) ae + py2" + p92? Seer a tre (oo) Wage (hai hee + Ys. (9°31) These will be called the partition functions f of the types of * Cf. Gibbs’ ‘Statistical Mechanics, p. 78. But (8°8) is generally valid, while Gibbs’ formula really refer only to a group of systems in a temperature bath. tT They are practically the ‘‘ Zustandsumme” of Planck, ‘ Radiation Theory,’ p. 127. 470 Messrs. C. G. Darwin and R. H. Fowler on system A and B. The application of the multinomial theorem then leads to the consideration of the expression [FE l9@T"s and pursuing exactly the same course as in § 5, we find | en VOMU@T . - > @8 ~ Qari =; | Safe orl wor. @9) Assume for the moment that we can choose a (2) con- forming to the requirements of §6. The whole calculation _ then goes on as before. The radius of the circle to be taken as contour is given by the equation B=Ms 4 log/(3) +N9 Z log g(3). . - (96) This equation has one and only one root. We thus can at once put down =[f(8)]-" 3, 1 5 LA8)]": ; (9°7) =Ms 5 log f(S). In exactly the same way we have a, = Mp 3/73), 0 00) Dn ee and we can also verify that in the case of an infinite bath the fluctuations are again given by (8°4), (8°5), and that equation (6°6) is still true. ‘The exact forms of the fluctua- tions (8°6), (8°7) are aiso valid if we replace re by e,. e have now to examine whether ¢(z) can be properly chosen. It is natural to take : o(2)=e Efe) [gO . . . (9°81) By its definition it must satisfy (i.). For (ii.) to be true, we must have E>Me.+ Na, which is the trivial condition that there must be enough energy to provide each system with the Jeast amount of energy it is permitted to have. Condition (iv.) does not appear at first sight inevitable, but must follow from Bohr’s Correspondence Principle *, for the convergence of the series f(z) and g(z) depends on their later terms—that is, those of * Bohr, doe. cit. the Partition of Energy. AT1 large quantum numbers. Condition (v.) is satisfied if not all the e’s and m’s have a common factor. There remains (iii.), and here there are trivial analytical difficulties when, as in general, M/E and N/E are fractional. B It is, however, easy to generalize § 6 by replacing [#(:) | by ~Er ¢7.\7Mr 474) & L/(2)] Lot) ] 2 and letting EK, M, N all tend to infinity independently. Condition (iii.) is then satisfied, as can be seen by multiplying out, and so all the conditions are satisfied, and the final results stated above are unaffected. Finally, we may observe that all our results can be extended at once to an assembly containing any number of types of system. ‘If there are M, systems of type C, for which the partition function is £,(3), then - d B= M$ clog f.(9), where $ is determined by d The formal validity of the proof will require all the quantities e, to be commensurable. It will be shown in § 12 how this restriction may be removed. § 10. Vibrators of two and three degrees of freedom. As a first example we take a set of vibrators each of which is free to vibrate ina plane under a central force proportional to the distance. The sequence of energies is again 0, e, 2e, ..., but the weights are no longer unity, as the system is degenerate. Following the principle laid down in § 2, we may evaluate the weights by treating the system as non-de- generate and counting the number of different motions which have the same total quantum number. «Now we can quantize the plane vibrators in directions w and y, and as an example for the case 2e, we have three alternatives (2e, 0), (ce, €), (0, 2c). This is easily generalized, and gives to re the weight r+1. The partition function for these vibrators is thus : fc) =14+ 27432? 4427 +..., Ed 472 Messrs. C. G. Darwin and R. H. Fowler on From the general theorem (9°7) we at once have = 2Me A Sele so that such vibrators have just twice as much energy as the line vibrators. In exactly the same way we can treat the case of three dimensions. To illustrate the weights we again take the case of 2e and quantize the system in w, y,z. There are six alternatives (2e, 0,0), (0, 2e,0), (0, 0, 2e), (e, €, 0), (e, 0, €), (0, €, e). The general form for ve is 4(r+1)(r+2). The partition function is now Ke) =14+324 627+ 1027+..., = (1-2-4, which leads at once to the expected result = 3Me WS gee § 11. Rotating Molecules. Another interesting example to which the calculations at once apply is that part of the specific heat of a gas due to the rotations of the molecules. Various writers* have quantized the motions of a rigid body, and it is found that the system has at most two instead of three periods, so that it is partly degenerate. We may consider for simplicity a diatomic molecule. Then, on account of the small moment of inertia about the line of centres, the third degree of freedom may be omitted altogether—its quantum of energy is too large. A simple calculation then leads to energies of rotation €. given by h? = Sr"? ay) Watch me neu nn pro Ut hve @bit) where I is the moment of inertia about a transverse axis, which we shall assume to be independent of vr. This is a degenerate system, and considerations of the number of cases which occur if it is quantized for the two degrees shows that the weight to be attached is 2r+1. This is on the principles suggested by Bohr f with a simplifying modifica- tion; for Bohr had to suppose that certain quantized motions were excluded for other reasons which are not operative * Among others, Khrenfest, Verh. Deutsch. Phys. Ges. xv. p. 401 (1913). Epstein, Phys. Zeitschr. xx. p. 289 (1919). F. Reiche, Ann. der Physik, liv. p. 421 (1917). t Loe, cit. p. 26. ee the Partition of Energy. 473 here *. There can, we think, be no question as to the correct- ness us the weight 27+ 1, put most recent writers have used a factor x; our formula for the specific heat has therefore a rather different value. We may now apply our general formule to this case with he ies WL Wo.§) ste ze : p,=2r+l1, ¢,=9r'e, e= 331° (11-2) Then {a= 1 ae a ee (LSS. E, =Ms, “0g f(3). ee cee aE ST) The contribution of i rotations to the molecular specific heat, C.,,, is dH,/dT, where M must be taken as the number of molecules in one gramme-molecule of gas. Thus, using (7°1), we have he deat d e : C= Tege eo lo g f(s Df : : (11 4) and Mk=R, the usual gas constant. If we write he ~ 8 Lk” uP) then d? C,4¢=Ro? + log (1+ 3e Sawer Men te (lot) Equation (11°5) shows that, when [>«,oa->0. It can be shown by the application of standard theorems on series f that when o—>0, ies ee teenta Piers NU) wapes ABE) which is the correct limiting value as required by classical dynamics. In the general case of any body we have three degrees of rotational freedom, the motion is simply degenerate f, and the energy enters as a sum of square numbers multi- plying two units of energy. The motion of the axis of symmetry and the motion about the axis of symmetry are not independent, and it is impossible therefore for the parti- tion function to split up into the product of two partition functions which represent the separate contributions of the two motions. The result is a double series of the same general type as (11°3). * Assuming that no extraneous considerations rule out any of these states, + Bromwich, Infinite Series, p. 132. The theorem is due to Cesaro. t Epstein, Phys. Zeit xx. p. 289 (1919). 474 Messrs. C. G. Darwin and R. H. Fowler on It does not appear profitable to examine these expressions further here, since the agreement with experiment is not very good at all temperatures. It is to be presumed that the assumption of constant moments of inertia is at fault, and this is supported by some of the evidence from band spectra ; further, it is probable that the case of no rotation must be excluded, involving the omission of the first term in the partition functions. The discussion of the practical applications of these formulee cannot be entered into here. $12. Assemblies containing free molecules. The problems we have so far discussed have all possessed the distinguishing characteristic that the temperature is the only independent variable. As soon as we treat of free molecules this is no longer the case, for now the volume must be another independent variable. Nevertheless, as we shall see, the same methods of calculation are available. The partition is no longer represented exactly by integrals, as it was for the quantized motions, but from the nature of the case some form of limiting process is required. The free molecules cannot of course be regarded as the limit of three-dimensional vibrators of low frequency, for they have no potential energy to share in the partition. We must proceed by the method common to most discussions of the distribution laws of classical assemblies—divide up into cells the six-dimensional space in which the state of any molecule is represented, associate with each cell a certain constant value of the energy, and in the limit make all the dimensions of all the cells tend to zero*. We take an assembly composed of M systems of the type A of § 9 and P free-moving monatomic molecules of mass m and of small size, the whole enclosed in a vessel of volume V. The energy of the molecules is solely their energy of trans- lation; they are supposed to obey the laws of. classical mechanics (except during their collisions with the A’s). In order to specify the state of the assembly, we™take a six- dimensional space of co-ordinates 91, g2,+++ 3, the three rectangular co-ordinates and momenta of a molecule in the vessel. We divide up this space into small cells, 1, 2, 3,..., ¢..., of extensions (dq,...dp3); which may or may not be * That the limit of the distribution laws worked out for the cells is the true distribution law for the actual assembly is an assumption implicit in all such discussions. the Partition of Energy. A75 equal. Then by the principles of § 2 the weight factor for the ¢th cell is = ee ay Mee i ah aCk) h provided of course that the cell is relevant to our assembly. Only those cells have a weight for which the q’s lie in the vessel; but the p’s may range over all values from —« to +a, for the method of summation will automatically exclude values which could not be allowed. Associated with the ¢th cell there is energy given by 1 e ! C.= te, (py + po? + p37). Renae pili (i227 bl) The state of the molecules in the assembly is specified by the numbers ¢, c., ... of molecules in cells 1,2,.... The specification of the A’sisas before. The number of weighted eomplexions corresponding to the specification is then APN ocrinn, pak . Ge ee RLS where 2 aoe > ae oc. th...” | (12°13) (ame Pa See et (1292) In proceeding thus we are constructing an artificial assembly in which the energy is taken to have the same value €, in all parts of the ¢th cell, and in which all the @s and all the e’s can be expressed as multiples of some basal unit, without a factor common to them all. This assembly can be made to resemble the real one to any standard of approximation required. For such an artificial assembly we can make use of the whole of our machinery. ‘The results all depend on integrals such as =a \anlOMB@r, . . 022) Qart where the partition function of the artificial molecules is Ree ead: Wheiyet m)s.(12-21) and the formule of § 8 follow at once for Ey, (Eyg—EH,)’, ¢ and (c,—¢,)”. These results give completely the exact partition laws for any artificial assembly of the type considered. To obtain the actual distribution law for the real assembly, we must make all the dimensions of all the cells tend to zero, and obtain the limit of the partition function. Now, by the 476 Messrs. C. G. Darwin and R. H. Fowler on definition of an integral, in the limit A(3)->H(S), where 1(© _ 10818 H@)= 2) gam PY Pe dg * dp, ae The integration is over the volume V and over all values of the p’s from —o to +0. This gives at once (27m)3??V Plog i/s)\72 In the formule the functions dh/d3 and d?h/d% also occur, and it is easily shown directly that their limits are dH/ds and d’H/dS°. We may therefore use (12°4) throughout for the real assembly, and at once obtain the following ex- pressions : H(s)= (12-4) E.= bas _ sper, (12:5) Teas RE CH. | eM Wiarton ee: (ey 3/2 1 j 9 1) C= =< S28) Dee ee aa ) dx dy dz du dv dw, » ((T2eon) 3/2 m ‘ = Sea eae On dw. eae The temperature 3 is determined by 13 qo « 3 i ). E= Ms J log 7(S)+$P Rese (do sa0) and the fluctuation of energy of the molecules in a bath of temperature 3 is given by (Bp—E,jind SE,=3PeT . . (1254) If the fluctuation of ¢, is evaluated, it takes the simple value (Gi 6) 6s «i a et seu, eae lee whether it is in a bath or not; for the second factor, analogous to that in (8°4), can be omitted when the cell is © taken to be of small size. Thus in all cases the possession of ¢,1s a normal property of the assembly. These results can be readily extended to cases where there is an external — field of force acting on the molecules. By means of this assembly we can establish the meaning of 3 in terms of T, by observing that the gas itself constitutes a constant volume gas thermometer. It is easy to show that the pressure of a gas must be $ of the mean kinetic energy the Partition of Energy. ATT in unit volume, that is to say, p=P/V log (1/3). Since the gas temperature is measured ; the relation pV =PAT, we are again led to the relation 3=e7"", We may observe that it is now possible to drop the assumption of commensurability, which was necessary in the sections which dealt with quantized systems. It was there essential, physically speaking, in order that it should be possible that the whole of the energy should be held somewhere; but as we now have molecules which can hold energy in any amounts, it may be dispensed with, the modifi- cation being justified on the same assumptions. and by the same sort of limiting process as have been used in this section. Again, we can see that the correct results are obtained if H(<) replaces h(z) in (12°2) and all the other integrals, even though the interpretation as coefficient in a power series is no longer possible, and though the integrand is no longer single valued. In such many-valued integrands the limiting process shows that we simply require to take that value which is real on the positive side of the real axis. §13. The Mazwell Distribution Law. We have carried out the whole process so tar with quantized systems included in the assembly, but it may be observed that it is immediately applicable to an assembly composed solely of molecules. If this is done the value of ¢; in (12°51) establishes at once the Maxwell distribution law, and its. fluctuation in (12°55) proves that it is a normal property of the assembly. This is probably the simplest complete proof of the ordinary distribution law ; its special advantage is that by means of the fluctuations it is easily established that the actual distribution will hardly ever be far from the average. The method can also be made to establish the distribution law for a mixture of gases*, and indeed for a mixture of any kind, provided that the systems can be considered to have separate energies. It is also possible to extend the method to cases in which the total momentum or angular momentum is conserved, by constructing partition functions in more than one independent variable. In fact, there will be as many independent variables as there are uniform integrals of the dynamical equations of the assembly. For simplicity we shall suppose that the linear momentum in a given direction is conserved, * The effects of the semi-permeable membranes of thermodynamics. can be conveniently treated by the partition function. 478 On the Partition of Energy. and let its total amount be G. The method now requires the averaging process to be applied to expressions depending on P! c,! Cate CHOC Oj Oo es where we now have not only Lic, =P, 70,510, but also 2 C= Ge where #, is the momentum in the given direction of a mole- cule in the tth cell. To sum the appropriate expressions we must take as our partition function hii i)\== Dd 25ta"t, With this function C will be the coefficient of 22° in [A(c,v)]*, and this can be expressed as a double contour integral. So can the other averages, and the usual asymptotic expansions can be found. ‘The correct distribution law follows on replacing h(z, x) by the integral which is its limit when the sizes of the cells tend to zero. This subject lies rather outside the theme of the present paper and need not be elaborated further. | $16. Summary. The whole paper is concerned with a method of calculating partitions of energy by replacing the usual calculation, whieh obtains the most probable state, and is mathematicaily un- satisfactory, by a calculation of the average state, which is the quantity that is actually required and which can be found with rigour by the use of the multinomial theorem together with a certain theorem in complex variable theory. 1 After a review of principles and two preliminary examples the real point of the method is illustrated in §5. Here there are two groups of interacting Planck vibrators of different types. It is shown that the partition can be found by evaluating the coefficient of a certain power of z in an expression which is the product of power series in z. This coefficient can be expressed as a contour integral and can be evaluated by a well-known method, the “‘ method of steepest descents.” The result expresses itself naturally in terms of a parameter 3 which is identified with temperature measured on a scale given by $=e7™, The work is extended to cover the partition among more The Heterodyne Beat Method. 479 general quantized systems in $9, and examples are given. In $12 it is shown how it may be made to deal with as- semblies composed partly of free molecules and partly of quantized systems. In $13 we deal with extensions possible when only molecules are present. The methods we have described can also be made to throw an interesting light on the statistical foundations of thermo- dynamics ; Gat in that connexion many points have arisen which require rather careful discussion, and in order not to make the present paper too long, we have deferred them to a future communication. Cambridge, May, 1922. XLV. The Heterodyne Beat Method and some Applications to Physical Measurements. By Maurice H. Bruz, M.Sc. (Cantab.), Barker Graduate Scholar of the University of Sydney * N a recent paper f, a preliminary account was given of the application of the heterodyne beat method to the measurement of magnetic susceptibilities. In virtue of the importance of the method as a sensitive measure of physical quantities, it seems desirable to give a more complete account of the principle and of some of the difficulties encountered in its application. Hssentially the method consists of the following arrange- ment shown in fig. 1. Two oscillating circuits, Set 1 and Set 2, are set up side by side and ar ranged so as to have approximately the same frequency. The two sets are loosely coupled so that in the telephone included in one of the circuits a resultant beat frequency is maintained equal to the difference between the frequencies of the fundamentals or overtones in the two circuits. If symmetry in the two circuits is essential, direct coupling can be replaced by indirect coupling by means of a third circuit in which the telephone is placed. In either case, when the beat frequency is low enough, an audible note will be heard in the telephone, and any changes in the constants of either circuit will cause the frequency of the audible note to alter by an amount equal to the change in frequency of the responsible circuit. This at once provides * Communicated by Professor Sir E, Rutherford, F.R.S. + Belz, Proc. Camb. Phil. Soe. vol. xxi. part 2 (1922). 480 Mr. M. H. Belz on the Heterodyne Beat Method avery sensitive method. It is now easily possible to maintain oscillations of frequencies up to 10’ per second. ‘Taking the case when Set 1 has a frequency of 1,000,000 per second, Set 2 a frequency of 1,001,000 per second, the audible note will have a frequency of 1,000 per second. If the frequency of Set 1 is changed to 1,000,001 per second, the frequency of the audible note will now be 999 per second, and this change in pitch can readily be observed by comparison with a note of standard pitch. Bigs yee all) 7 een Sena com See This sensitive method has been successfully employed by Herweg *, Whiddington ft, Pungs and Preuner ft, Falcken- burg §, and several others in physical researches, but the precautions necessary for steadiness in the beat note have never been completely specified. Precautions. With high frequency oscillations of the order 3 x 10° per second to 5 x 10° per second such as were used in the present investigations, electrostatic shielding from all external in- fluence was of the first importance. This was ensured by placing all the elements of the circuits in earthed metal- lined boxes, one of the variable capacities, by means of which final small adjustments were made, being provided with a long ebonite spindle which projected beyond the containing box. With the box closed the note from the telephone IT was considerably reduced in intensity, and in order to obtain the maximum loudness, a small section was removed from the box, shielding being maintained by means of a piece of fine metal gauze. * Herweg, Zeit. f. Phys. vol. 111. p. 86 (1920). + Whiddington, Phil. Mag. vol. xi. p. 634 (1920). t Pungs and Preuner, Phys. Zeit. vol. xx. p. 543 (1919), § Falckenburg, Ann. d. Phys. vol. |xi. 2, p. 167 (1920). and some Applications to Physical Measurements. 481 Solidity of foundation is a most important requirement. In the experiments of Whiddington*, although the apparatus was set up on a solid base, vibrations of the building even at 2 a.M. proved troublesome. A somewhat similar trouble was experienced in some of the earlier experiments when the apparatus was installed on the top floor of the laboratory. It was found that the vibration of the building caused by people walking about the corridors, and by the passage of heavy motor traffic, appreciably affected the steadiness of the note. Although some of the work was done during the night and over the week end, the trouble always persisted. Finally the apparatus was transferred to a room on the ground floor and supported on stone pillars by means of solid rubber pads. The trouble was now completely removed so that successful observations could be made during the day despite the fact that people were continually walking beside the apparatus. After lighting the valves, a certain amount of time must elapse before the oscillating system has settled down to a steady state. This initial variation is due to the heating and expansion of the elements of the valves, causing changes in the whole capacity linked with the oscillating systems. In order to save time, thus sparing the high tension batteries and prolonging the life of the valves, the latter were contained in small tin boxes, placed outside the large box, which were lagged with asbestos and cotton-wool. In this way the heat conduction was minimized, and the valves settled down much more rapidly. Other conditions being the same, it was found that certain valves were less satisfactory than others. For some types the settling down process was very long, and by the time the valve was set, other things began to vary. After long trials with “ R,” “ Fotos,” and ‘“ A. T.” types, it was found that “B” type valves manufactured by the General Electric Company gave most satisfactory results, settling down most rapidly and remaining steadiest. The effects of the changes in the elements of an oscillating system on the frequency have beeu examined by Eccles and Vincent f in the case of wave-lengths of 3000 metres. They determined that between certain limits for each value of the coupling between the plate and grid coils there was a particular value of the filament current for which the wave- length was a maximum. Working at this value of the current it was found possible to hold the beat note steady to * Whiddington, loc. cit. + Eccles and Vincent, Proc. Roy. Soc. A. vol. xcvi. p. 455 (1919). Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. ria | 482 Mr. M. H. Belz on the Heterodyne Beat Method one part in 100,000 for several minutes in spite of small unavoidable variations. With the frequencies employed in the present experiments, however, such a condition could not be established. The heating of the valve parts and the consequent change in capacity in the system resulting from changes in the filament current cause changes in wave-length which certainly far outweigh any real change due to increased thermionic emission alone. In order then to secure a constant filament current, accumulators of 100 ampere-hours capacity were employed. These were charged regularly after about three days’ use, and after the valve had settled down, the current from them showed no variation during a single run. Faulty contacts of wires joining the elements of the circuits were avoided by soldering, the only sources of uncertainty being the sliding resistances in the filament circuits. These, however, were good types with bright surfaces and stiff springs so that the chance of error due to change of contact was small. The principal cause of variation in the frequencies of the circuits was found to be due to variations in the high tension batteries. This trouble has been mentioned by Hecles and Vincent *. In the present work the plate voltage was obtained from trays of portable accumulators of fairly low capacity, each tray providing 40 volts. After the valves had been burning for an hour or so, taking a current of about 10 milliamperes, this voltage began to vary and the beat note consequently drifted. However, giving the valves time to settle down, a matter of 15 to 20 minutes, it was found possible to hold the heterodyne note quite steady for intervals of 30 to 60 seconds, and this is ample time in which to make a single observation. After about 90 minutes burning the variation was too rapid and the batteries had to be recharged. The size and consequent capacity of the cells of these batteries is limited by the fact that they have to be contained in a metal box, and thus this source of variation ean only be provided for in special cases. Technique. It is essential to maintain the oscillations generated in the circuits at frequencies considerably different trom the natural frequencies of the coils alone, that is to say with an appreciable capacity in the system, and under these conditions the frequency, n, of the oscillations in such a circuit containing * Eecles and Vincent, doe. cit. + Cf. Townsend, Phil. Mag. vol. xlii. August (1921). and some Applications to Physical Measurements. 483 inductance L and capacity C, is given very approximately by n=1/(27,/LV). Changes in » can thus be brought about by changes in ( or L. In the experiments to be described below, the changes in n were brought about by variations in L, and in this case, with C constant, a small variation, dL, in the inductance produces a corresponding change, dn, in the frequency given by Ory SMa oem iy ay fy ael me (L, ) The experimental part thus reduces itself to a determination of dn. This is accomplished by obtaining beats between the heterodyne note and a note of constant pitch, and then counting the change in the number of beats per second caused by the change ininductance. A. considerable amount of practice in listening is required in order readily to be able to adjust the heterodyne note to the pitch of the constant note. This note can be very conveniently obtained by means of a third set, some distance away from the other sets, oscillating with audible frequency, in the plate circuit of which a telephone is placed. The intensity of the note heard can be altered by adjusting the filament current, and in this respect the note is very much more convenient than that obtained from a tuning-fork. For it was found that the heterodyne note could be more easily brought to tune with the standard note, and false beats more readily recognized when this latter could be altered so that both notes had approximately the same intensity. In some of the experiments * it was found impossible to obtain a beat note of convenient audible frequency when the fundamental frequencies of the oscillations were approxi- mately the same. It was observed that, as the capacity of | one of the sets was altered, only very shrill notes could he heard on either side of the very large region of silence. This synchronization effect appears to depend on several factors, but chiefly on the coupling between the circuits. Cn account of the limited size of the box containing the coils, the coupling could not be reduced beyond a certain lower limit, and reducing the strengths of the oscillatory currents merely reduced the intensity of the limited note. In ali these cases it was possible to obtain the heterodyne note between the fundamental of Set 2 and the first overtone of Set 1, which was quite steady and possessed the normal region of silence. Since the changes in 7 were produced by * Those in which the determinations of the magnetic susceptibilities of certain salts were made, see below. 2.1 2 484 Mr, M.H. Belz on the Heterodyne Beat Method variations in the inductance of Set 1, this arrangement increases the sensitiveness of the method. For let N be the frequency, determined at the centre of the region of silence, of the fundamental oscillation in Set 2, the Trequency of the fundamental oscillation in Set J, then since the tirst overtone of Set 1 is employed to produce the note, N=2n. Let the frequency of the audible note from the third circuit be m. Then when the heterodyne note is adjusted, by slightly varying the capacity of Set 2, so that q beats per second are counted, the frequency of Set 2 is (N4+m2q) =(2n+m+q). If now the frequency of the fundamental oscillation of Set 1 is altered by dn per second, the frequency of the first overtone is altered by an amount 2dn per second, so that the frequency of the heterodyne note is now (2n+m+q)—(2n+42dn)=(m+q+2dn), whence if a change of p beats per second is observed when the induc- tance change is accomplished, p=2dn. The sensitiveness is thus doubled, and could similarly be increased by em- ploying higher overtones of Set 1. Against this, however, is the fact that the notes so obtained are very feeble, and counting becomes increasingly difficult. From equation (i.) we see that the sensitiveness depends onn. Itis now possible to maintain oscillations of frequencies up to 10‘ per second, but in cases where the change in in- ductance is caused by inserting a specimen within the coil, there is an upper limit to determined by the form and function of the coil L. It is necessary to divide this coil into two parts between which there is no mutual inductance, one part L, being coupled to the grid circuit in order to maintain the oscillations, the other part Ly serving as the coil in which the inductance changes occur. This latter part must be a fairly long coil in order that there may be an appreciable region within it through which the magnetic field is constant, in which region the specimen is placed. On account of the dimensions of this coil, the first part has to possess a fairly large inductance in order to get sufficient mutual inductance with the grid coil: further, a certain amount of coupling is required with Set 2 to produce the heterodyne note. Haperimental. In the present experimental arrangements the details of the coils are as follows :— Coil 10pe The coil was 10 cm. long, and consisted of 100 turns of copper wire, no. 22 s.w.g., double cotton covered. It was and some Applications to Physical Measurements. 485 wound on a short length of glass tubing and had an effective diameter of 2°10 cm. The self-inductance, employing the exact formula of Nagaoka™*, viz., where L, is the self-inductance of a current sheet of the same dimensions as the coil, n,; the number of turns per em., a the effective radius, 6 the total length, and K a factor depending on the ratio of the diameter of the coil to the length, to which was applied the correction for spacing, was calculated to be 39,160 em. The small frequency correction was nevlected. Coil Le. The coil was 30°70 cm. long. and consisted of 541 turns of copper wire, no. 24 s.w.g., silk covered. It was wound on a long glass tube, of external diameter 1°00 cm., and separated therefrom by means of a layer of paraftined paper. The effective diameter (2a) of the coil was 1:105 cm., and self-inductance, calculated as above, was 92,430 cm. The total inductance L (= L,+ I) is thus 131,600 em. The coil L, was outside the box containing the rest of the circuits, and was shielded from external electrostatic influences by means of an enveloping earthed metal cylinder. Coil L;. The length was 9 cm., and the coil consisted of 90 turns of copper wire, no. 22 s.w.g., double cotton covered, It was wound on a short length of glass tubing and had an effective diameter 4:13 cm. ‘The self-inductance was similarly cal- culated to be 124,600 cm. The capacities employed had a range of 100 to 1200 microfarads and were provided with a slow movement. Changes in the frequency of the oscillations of Set 1 brought about by the insertion of a specimen within the coil L, may be due to three causes :— (a) In the first place, if the coil is not shielded from the electrostatic effect of the specimen, the self capacity of the coil will be changed. In order to observe changes of inductance alone, it is necessary to guard against this possibility. This was done by depositing a thin layer of platinum ¢ on the outside of the glass tube on which the coil L. was wound, and earthing. The thickness of the deposit, obtained by weighing, was 7x107-& cm. It is necessary to * Nagaoka, Jour. Coll. Sci. Tokyo, xxvii. art. 6, p. 18 (1909). + The function of the paraffined paper was to prevent any possible short-circuiting of the coil through the layer of platinum. 486 Mr. M. Gi. Belz on the Heterodyne Beat Method determine the effect of this shield on the strength of the magnetic field within. The magnetic force, H;, at a depth ¢ in a mass of metal is related to the force, Hy, at the surface by the equation 2arupt Tone ow cor. in which pw, o represent the permeability and_ specific resistance respectively of the metal, and p=2n, n being the frequency. Taking n=4°84 x 10° per second, the largest frequency used, and for platinum, w=1, c=11,000 c.g.s. e.m.u., we find that within the shield for t=7 x 10° em., BL EL) =e, ON = 0°9997, The effect of the field on the changes of inductance can thus be neglected. | - With the shield, the remaining causes of the change of frequency are due entirely to changes of inductance, being (6) an eddy current effect within the specimen, and (c) in the case of the magnetic substances, a susceptibility efrect. The Eddy Current Effect. The magnetic field, H, within the coil will be of a harmonic type, and on this account circular eddy currents will be induced in the specimen in planes perpendicular to the axis of the coil and in sucha direction that the magnetic forces arising from them oppose and consequently diminish the value of H. This virtually means a diminution in the inductance of the coil, and the frequency change will be in the direction of nm increasing. ‘This result can also be predicted mathematically by regarding the specimen as equivalent to a coil with self-inductance and resistance, coupled to the main coil. The analysis, however, is com- plicated and its development was not proceeded with. Some experiments were made, however, to determine the order of the change in inductance and its dependence on the charac- teristics of the specimen. In the first place, as bearing on the results obtained in the magnetic measurements, sulphuric acid was examined. This was contained in a long glass tube, the effect of which had previously been determined to be zero, and was lowered into the oscillating coil L,. Any change in inductance due to an eddy current effect should certainly depend on the and some Applications to Physical Measurements. 487 conductivity of the specimen. This solution had a con- ductivity far greater than any of the magnetic specimens, which were examined in liquid, crystal, and powder forms, but, counting over periods ranging beyond ten seconds, no variation in the beat note was observed. Accordingly it was assumed that the effect in the magnetic substances was negligible. Some tests were then made with different lengths and sizes of graphite taken from ordinary pencils. The con- ductivities of these specimens were much greater than that of the solution of acid, but there were no changes in the beat note indicating an increase of frequency. In the case of some specimens changes occurred which correspouded to an increase of inductance, but some iron must have been present in them. Finally, some tests were made on ditferant specimens of copper, brass, manganin, platinoid, and constantan wires. In these experiments, which were carried out after the susceptibility determinations, some alterations in the elements of the circuits enabled the heterodyne note to be obtained by beats produced between the fundamental oscillations. It is interesting, at this stage, to give the results obtained for the variation in frequency change with the conductivities of the specimens. In this connexion two tests were made, the first employing small specimens and counting the beats, the second employing large specimens and computing the frequency change from the alteration in capacity necessary to bring the heterodyne note to coincidence with the auxiliary note. (a) The specimens used were wires of copper, brass, platinoid, and manganin, each of length 0°32 cm., and diameter 0'711 mm. The wave-length was 710 metres. The results are included in Table I. TaBLe LI. Frequency Change Specimen. per second. dn. BUAMIICES oe fay diss cad Bond eckh an oh 9°54 PSO yeas Like Ree SK encek ds o> 4°88 DN RONIN Rera she a0 nce x20 0°33 LN ae ee 0:28 (b) The specimens used were wires of copper, brass, and constantan, each of length 8 em. and diameter 2°03 mm. 488 Mr. M. H. Belz on the Heterodyne Beat Method The wave-length was 725 metres. The results are included in Table II. Tasze II. Capacity Change Frequency Change Specimen. ne as per second. (arbitrary units). Fs Copper ists devas ee eeeee 201 1027 EB ASS CEPA E wie. ok tame Sho 190 961 Monstamtan oka essa ne tones 98 501 The wires of platinoid, manganin, and constantan used in the above tests contained 15 per cent., 4 per cent., and 40 per cent. nickel respectively. The magnetic suscepti- bilities of copper-nickel alloys have been determined by Gans and Forseca*, who found that when the nickel content was 40°4 per cent. the susceptibility was only 0°189x107-® cg.s., being much smaller for lower per- centages. The effect of the susceptibilities of the specimens used on the frequency changes can thus be neglected. Fig. 2. oCurve B nN ° COIS. —— > frequency change, 07, pers ES ° to) SB o 0 Oxio> 1 2 3 4 5 6x 10° ——> Specific Conductivity, KR, mhos. ¢cn.-’. The specific conductivities, K, in mhos cm7}., of the materials used, taken, with the exception of brass, from Kaye and Laby’s tables, are:—for copper, K=6-29 x 10°, for platinoid, K=0:291 x 10°, for manganin, K=0-233 x 10°, and for constantan, K=0:204x 10°. The specific conduc- tivity for the brass used was determined to be K =1'182 x 10°. Mica? represents the relations obtained by plotting dn against K, curve (a) referring to test (a), curve (0) to test (0). * Gans u. Forseca, Ann. d. Phys. vol. lxi. p. 742 (1920). and some Applications to Physical Measurements. 489 The forms of both curves indicate a relation of the type K2 dlL=constant x K?4B° the term B varying for different sizes of specimen, being relatively less important the greater the whole conductivity of the specimen, as shown in curve (b). A similar type of dependence was calculated by Mr. Kapitza, of the Cavendish Laboratory, for the case of a thin spherical shell. For solid cylinders, the dependence of the frequency change on other characteristics of the specimens was examined, but it is hoped to give a complete account of the results together with further tests on different forms of specimens, in a subsequent paper. THE APPLICATION OF THE METHOD TO THE DETERMINATION OF THE MAGNETIC SUSCEPTIBILITIES OF CERTAIN SALTS. The general principle of the method has been given in the previous paper*. It has since been applied to the determination of the magnetic susceptibilities of many salts, the results obtained being given below. In addition it seems desirable to indicate more carefully the nature and magnitude of the corrections to be applied. Considering the insertion of a magnetic substance in the form of a cylinder of cross section A’ and length l', the volume susceptibility of which is K,, within the coil L of Set 1, the cross section and length of which are A and / respectively, thus causing the frequency, n, of the oscillations to alter by an amount dn, we find, subject to the corrections to be given below, K i dn Al L, ae L, : sider Wa ae Wn Wn OS In all the experiments for determining K,, the heterodyne note was obtained by beats between the first overtone of the oscillations in Set 1 and the fundamental oscillation of Set 2, as already mentioned on pages 483, 484 supra. In this cease, if N is the frequency of the oscillations in Set 2, and a change of p beats per second is observed on inserting the specimen, N=2n, and p=2dn. We thus obtain an 1 P Al I+ L, ee — dN AT i- ° ° ° e (11.) * Belz, loc. cit. 490 Mr. M. H. Belz on the Heterodyne Beat Method Before this formula can be applied it is necessary to— investigate several corrections. The absorption in the platinum shield which covered the tube on which L, was wound, and the eddy current effect, have been discussed and shown to produce no correction terms. But there remain two further corrections to be investigated, (i.) due to the demagnetizing effect of the specimen, and (ii.) due to the end correction term in the calculation of the self-inductance of the coil and the effect on this due to the position of the specimen. (i.) The demagnetizing effect on the magnetic field due to the magnetism induced in the specimen. By treating the specimen as an elongated ellipsoid of revolution, of semi-minor and semi-major axes @ and ¢ respectively, with its long axis parallel to the field, Ewing * has shown that if H is the value of the magnetic force. within the specimen, and H’ the original value of the magnetic force before the specimen was inserted, then H=H’'(1—NK;), where N is given by Ne4r(l/e—1) J © slog. (1 +ey/(1—e—1}, e being the eccentricity and equal to ,/1—a?/c’. In a typical ease, that of cobalt chloride in solution, which had a value of Ky, approximately 20x 107° c.g.s., 2a=0°507 c.m., 2e=8'0c.m. Hence e=0°998. This gives N=0'126, and hence H= H’(1—0°126 x 20 x 107°) = 292 x 101) so that this correction is negligible. To get the effect on the external field, consider the resultant magnetism induced on the ends and sides of the specimen. Ewing} shows that the free magnetism, although densest at the ends, extends towards the middle, and it is only on the equatorial line that there is none. Also the total quantity of free magnetism on any narrow zone taken perpendicular to the direction of magnetization * Ewing, ‘ Magnetic Induction in Iron and other Metals,’ pp. 23-25. + Ewing, loc. cit. pp. 25-27. and some Applications to Physical Measurements. 491 is proportional to the width of the zone and to its distance from the equatorial line. Consider such a zone at distance # from the equatorial line, x being measured in the positive direction of the field, width dx. Then the total quantity of free magnetism on it is equal to +Cxda, where © is determined from’ the fact that the total quantity of positive (or negative) magnetism, a. ef Cate is equal to zal, I being the intensity of magnetization. This gives C= rs ii and hence the free magnetism on the strip is + Qa yes wala. Let us consider the effect: of this quantity of positive magnetism at the distance x, and an equal quantity of negative magnetism at —# ata point on the equatorial plane outside the specimen. Suppose the free magnetism on the strip to act, in regard to external points, as if it were concentrated at its centre of gravity, and let d be the distance of the point in question from the axis of the specimen. Then the magnetic force in a direction parallel to the axis at the point 0, d will be a? ada Hh —_ FTN I s e —— — — 5 47 > a + dP + c and the effect due to the whole specimen will be the integral of this from 0 to c, which becomes —4n 3 ea ;1.(- Pers If H is the value of the verte force within the specimen, which we have seen is also the original value of the field within the coil before the specimen was inserted, we may write [=K,H, and the field at the point 0, d is reduced from H to 14a eee higinih - 1 H {1-4 ce Ae i Taz ae + sinh a Taking d=0°5 em., i. e. just within the winding of the coil, and with the same numerical data as before, this becomes H(1—1°79 x 10-8). Ata point on the surface of the specimen, for d=0°25 cm. we obtain for the reduced field, H(1—2°5 x 10-). The correction in this case is also negligible, and we are — at sini a). 492 Mr. M. H. Belz on the Heterodyne Beat Method thus justified in completely neglecting the effects of de- magnetization. | (i1.) The effect of the end correction term and the position of the specimen. In a long coil there is an appreciable length over which the magnetic force is constant and given by H=47ny, ny, being the number of turns per unit length. At the ends the force falls to half value, and this diminution in H is responsible for the correction term K in Nagaoka’s formula Jy aed) 6 i When this correction is small and so can be neglected, the expression (il.) becomes exact. In the present case, the value of K was equal to 09873 and the error involved in neglecting this end correction is about 1°3 per cent. We can allow for the correction in the following way : The shortspecimen (about cm. as compared with the length of the coil, 36°70 cm.) was suspended in approximately the centre of the coil, which can thus be supposed to consist of three coils in series, one in the centre of length equal to that of the specimen and without end correction, and two approxi- mately equal coils on either side, to the open ends of which the correction is to be applied. Let /’ be the length of the specimen, / the length of the coil, then the total self- inductance is L=C/K, while the self-inductance of the coil of length J’ is L,=Cl’, where C=4m’a’n,’. The area of cross section of the coil being A, that of the specimen A’, the self-inductance of this portion is altered to Ly’ where L,’=1L)(1+47rK,A‘/A), so that the whole change in inductance of the coil, dL, is given by du=1)47K,A'/A, = L)47K,A'l'/(AIK), and thus the expression (ii.) in the corrected form becomes m2 pep K .volume of coil L,+ L, Se , NS Gime 2a N volume of specimen’ Ly Gn EHeperimental Errors. The accuracy of the estimations depends almost entirely on the determinations of the change in the number of beats per second and of the frequencies. The counting was done by means of a stop-watch guaranteed to read tenths of a second. ‘lhe heterodyne note was adjusted by means of the fine movement on the condenser of Set 2 so that a reasonable and some Applications to Physical Measurements. 493 number of beats per second, sometimes zero, was heard. The time for 20 beats was noted, the specimen then inserted the new time for 20 beats being again observed, and finally a third count made when the specimen was removed. ‘The note was then slightly varied and the same procedure repeated. In this way from 6 to 10 readings were obtained for each particular frequency used. A review of all the measurements so made under conditions when the heterodyne note was steady show that the greatest error incurred was about 2°2 per cent. The wave-lengths were measured with a calibrated Towns- end wave-meter, and the resonance point could be fixed to within 2 metres. ‘he shortest wave-length measured was about 350 metres corresponding to a fundamental oscillation in Set 2, so that the maximum error involved here is about 0-6 per cent. From all sources, then, the maximum error involved in the estimation of the susceptibilities is about 3 per cent. The Experimental Results. The specimens were all prepared by Kahlbaum, Berlin, and were contained in short lengths of glass tubing thus permitting of quick insertion into the coil. The effect of the tubing alone was found to be zero in every case. The results are given below. I. Ferric Chloride—The salt was examined in the form of a solution in air-free water. It was contained in a closed glass tube, occupied a volume 1°321c.c.. and had an approxi- mate length 7°5 cm. The density was determined to be 1152 gm. per e.c. at 15° C., and a volumetric estimation showed that the solution contained 0:189 om. FeCl, per c.c. The observations are shown in Table III. Taste III. | Frequency | Frequency | Change in & pe re of Set 2 of Set 1 number of rade ae eee per sec. per sec. | beats persec.| p/N. Fae say CG in metres. N. Ns p=2dn. egrees C. x 10° Sc16? sein? O72 8:07 4:03 2°59 3°21 15 385 7°80 3°90 2°49 3°19 15 400 | 7°50 3°75 2°39 3°19 5) 418 - foe erin 259 2-29 3:19 15 Mean p/N=3'19x 107° at 15° O. 494 Mr. M. H. Belz on the Heterodyne Beat Method The values of p/N show that for the range of frequencies used the susceptibility is constant. This was found to be the case for all the substances examined. Formula (iii.) gives ° > \2 e : Kot 319 x 10-8 (7/4 CL 105)? x 36°70 x 09873 20 te a2 ,. 131600 = ILRI Sek s C.2.5S. 99420 The mass susceptibility of the solution K,,, is thus Kini LOK LO ne ail 2 — hoon eis For a solution of a salt in water, the mass susceptibility is given by Ip = pK, a5 @ —Po VK es where p)=mass of salt per c.c., K,,=mass susceptibility of the water-free salt, K,,,= susceptibility of water, which may be taken as -—0°75 x 107° c.g.s. We thus obtain = 9007 x LOT Sic och at ion C: The values for K,, determined previously by balance methods are* 92x10*, 91 «x10°°, 33x10°" O2Zegiias 88 x L0-8, and 103 x 10~¢ ¢.g.s. with a mean value 91 » 107 c.g.s. The present result is in good agreement, and thus shows that up to a frequency of 4:03 x 10° per second the susceptibility is not altered. II. Ferrous Sulphate-——Tests were made on the salt in forms of crystals and powder. The crystals were good ones of the monoclinic type, and on estimation proved to have the composition, FeSO, 7H,O almost exactly. Hqual masses of crystals and powder were used, the powder being obtained by grinding up the crystalsin a glass mortar. The substances were enclosed in glass tubes, the transference after weighing being rapid to avoid absorption of water from the atmosphere. The mass of each specimen was 1:00 gm. at 15°C. At this temperature the density of FeSO,, 7H,O crystals is 1°899 om. per c.c.,and hence the volume of each specimen was 0°527 c.c. In the form of powder this volume occupied a length 5°6 cm., while for the crystals the length was about 8 cm. * Landolt and Bernstein Zabellen (1912). and some Applications to Physical Measurements. 495 The observations are shown in Table LV. TABLE LV. | 3) F Cl i | | Brequency requency lange in ‘ 7 ie | of Set 2 of Set 1 number of yen? ee |: par sec. per sec. beats per sec. p/N. ee in metres. | N. n=N/2. p=2dn. | degrees C, _ Cryst. |Powder.| Cryst. | Powder. | AS eHOR at) oe 10° 10-5" | «107 352 Le: S58 L426. 4-40 4:40 515 SN Pana a eee | 4:00 4:10 | 412 5:13 ioe 9.16 392 | 765 3°83 3°91 3°90 SE 510 | 16 410 | 7°32 3°66 O74 371 5:11 5'07 16 . 414 beitceaitas | S63 3°70 3°69 510 509. -|.- 16 a. T20 1 VSO S002 bla us ts Be aN ey names a 16 Mean p/N for both erystals and powder=5'12x107® at 16° C. The susceptibility of the salt is thus the same in the solid as in the powder form. A similar identity in the sucepti- bilities of different forms of the same material was observed by Wilson * using a balance method in the investigation of certain iron ores. The volume susceptibility is K,=76°6 x 10-® ¢,9.5., whence the mass susceptibility of the complete salt is 40-4x10-* c.g.s. The percentage mass of FeSO, in one eram of salt is 54°6, so that, neglecting the contribution to the susceptibility of the water of crystallization, we find Ki, = 120 X10 7".¢.5,5. at 16°C. The mean of the values for K,», all determined in the form of a solution, given in the Tabellen of Landolt and Bornstein is 75x 107% c.g.s. Finke+ found for K, for the complete salt a value 80x 107° ¢.g.s., and the present result agrees very well with these. ; Ill. Ferrous Ammonium Sulphate.—The salt was examined in the form of monoclinic crystals which proved, on estima- tion, to have the composition FeSO,(NH,).SO,, 6H,O to an accuracy of 0°3 per cent. The density of this salt.is : 1°813 gm. per c.c. at 15°C. A mass of 1 gm, was taken and : placed in a glass tube, occupying a length of about 8 em., * Wilson, Proc. Roy. Soc. A. vol. xcvi. p. 429 (1919). 1 Finke, Ann. d. Phys. (4) xxxi. p. 149 (1910). / 496 Mr. M. H. Belz on the Heterodyne Beat Method and a volume 0°551 ec.e. We find hence 29°8 x 1076 c.g.s. The observations are shown in Mean p/N=3'79X10~® at 16°C. Ke 0450 x10 etic ers: which gives for the mass susceptibility of the complete salt in one gram of salt is 72°5, and hence we obtain pO Se LO eas, aie ILE Oe The values given in the Tabellen of Landolt and Bornstein Table V. ; TABLE V. Change in Wave-length| Frequency | Frequency S | Tempera- of Set 2 of Set 2. of Set l. number of Ae in metres beats per sec.| p/N. degrees C ; N. w= N72. p=2dn. 8 : 105 - x 10° «10-8 335 8:95 4:47 3°39 3719 16 340 8°83 4°41 3°34 3°78 16 363 8°72 4:13 ili 3°79 16 400 7:50 3°75 2°84 3°79 16 The percentage mass of FeSO,(NH,).SO, of K,, for the salt when examined in the form of a solution at 18°C are 44x 1076, and 45 x 10-6 c.g.s. IV. Mickel Chloride.—The salt was examined in the form of a solution in air-free water. The volume of the specimen was 1°572 c.c. and occupied a length in the glass tube of approximately 7-4cm. The density was determined to be 1°332 gm. per c.c., and an electrolytic estimation yielded 0°255 gm. NiCl, per c.c. The observations are shown in Table VI. Tasie V1. Wave-length| Frequency | Frequency See a Tempera- of Set 2 of Set 2. of Set 1. hears »/N ' ture i tres. SEN Dem ees TS) degrees C. Tae IN n=N/2. p=2dn. @ x 10° x 10° Same , 309 8:45 4:22 2°12 2°31 15 380 7°90 3°95 1:99 2°52 15 404 7:43 371 1:88 2°52 15 Mean p/N=2'52X107 at 15°C. and some Applications to Physical Measurements, 497 _ This gives | ee bebe «10 el ps: The mass susceptibility of the solution is B= 949x107 c.g.s., and the mass susceptibility of the water-free salt becomes Ke pSod'o X:LOT? c.o-s) ab, 15°C. This is in good agreement with the values given in the Tabellen of Landolt and Bornstein, the mean of which is 39°66 x 1078, but is considerably different from the value recently given by Théodoridés *, viz. £47°6x107° c.g.s. at 15°-2 C. V. Nickel Sulphate—The specimen was in the form of the green rhombic prism crystals which, on estimation, were shown to have the composition NiSO,, 7H,O; the density of these crystals is (Thorpe and Watts) 1:950 gm. — perc.c. A mass of 1°438 gm. occupying a volume 0°737 c.c. was used, having a length in the glass tube of approximately 7°-2cm. The observations are shown in Table VII. TaBLeE VII. _ Wave-length| Frequency | Frequency |— ek od a Tempera- of Set 2 of Set 2. of Set 1. es N ture ; ance eats persec.| p/N. degrees O ae Ne n=N/2. p=2dn. x 105 x 105 alms 342 8°77 4°38 2°63 3°00 16 370 812 4°06 2°44 3°00 16 412 7:29 3°64 2°16 2°96 16 Mean p/N=2:99 x 10~° at 16°C. This gives | K»=32'1x 107° c.g.s., | from which we get the mass susceptibility of the complete salt to be 16°43 x 107%c.g.s. The percentage mass of NiSO, in one gram of salt is 55, and hence we obtain 29-9 x 10 Scre.s; at L6> ©. The mean of the values in the Jabellen is 30X10~° c.g.s., while Finke + finds for K, the value 29°1 x 10~$ c.g.s. * Théodoridés, /ouwrn. d. Phys. III. i. p. 1 (1922). t Finke, loc. cit. Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2K 498 Mr. M. H. Belz on the Heterodyne Beat Method VI. Cobalt Sulphate.—Tests were made with the salt in the form of crystals and powder. (a) Tests on crystals. The salt proved on estimation to have the composition CoSO,, 7H,O ; the density of this salt is 1°950 gm. per c.c. A mass of 1:1375 gm. was used which thus occupied a volume 0-583 c.c. The length of the crystal column was 6 cm. The observations are shown in Table VIII. TABLE VIII. Wave-length | Frequency | Frequency nee o Tempera- of Set 2 of Set 2. of Set 1. ture in metres beats per sec.) p/N. degrees C N. n=N/2. p=2dn. 8 x 10° x 10° %e.o-g: at 16° GC. For the complete salt Finke* finds K,=68 x 107° c.g.s. The mean values for K,, in the Tabellen is 66 x 10~¢ c.g.s., while Théodorideés } gives for K,, the value 64°5 x 10~®c.g.s. aE EG. VII. Cobalt Chioride.—The salt was examined in the form of a solution in air-free water. It occupied a volume 1°620 c.c. and had a length in the glass tube 8 em. The density was determined to be 1:186 om. per c.c., and an electrolytic estimation yielded 0°2078 gm. CoCl, per c.c. The observations are shown in Table X. EABILE X. eo Change in Wave-length| Frequency | Frequency Tempera- of Set2 | — of Set 2. of Set 1. | abe ture in metres. | 2 jbeats per sec.) p/N. degrees C N. n=N/2. p=2dn. 8 «10° x 10° S10" 340 8°82 4-4] 3°48 3°94 20 385 7°80 3°90 | 3:09 3°96 20 | 414 | 7°25 | 3°61 | 2°85 3°94 20 Mean p/N=3-95 x10—® at 20°C. This gives | E30 105 eres. from which we find for the mass susceptibility of the solution Km, = 16°28 x 10-® c.g.s., and finally for the mass susceptibility of the water-free salt ie dix 10—* e.a-s: ab) 20°C. * Finke, Joc. cit. + Théodoridés, loc. ctt. Fag | Sb 500 On the Heterodyne Beat Method. The values in the Tabellen for K,, are 81 x 1076, 101 x 108, and 82 x 10~&ce.g.s., while Théodoridés* gives 97°9 x 10-®c.g.s. The present result thus indicates that the lower values are the more accurate. Summary of Results. The results obtained are grouped together in Table XI. TaBLe XI. ; Form in Mass susceptibility Substance. sv hiok tesla: Sf ea Temp. C. Ferric Chloride, FeGl,.| Solution ...... 90°7 x 10—° ¢.g.8. 15° Ferrous Sulphate, Crystals and 74:0 x 107° ¢.g.8. 16° sem FeSO,,7H,O. Powder. Ferrous Ammonium §ul- | Crystals......... 41-1 x 1076 &g-8. 16° phate, FeSO,, (NH,), SO,, 6H,O. . Nickel Chloride, NiCl,.| Solution ...... 39:5 x 10-6 ¢-g.8. 15° Nickel Sulphate, Crystals 29-9 x 106 &-g-s. 1162 NisO,,7H,0. ; Cobalt Sulphate, Crystals and 61 910-6 «.g.s. 16° CoS8O,,7H,0. Powder. Cobalt Chloride, CoCl,.| Solution ...... 81:°3x 10-6 ¢.g:s. 22 Summary. The precautions necessary for steadinessin the heterodyne beat method are described, employing frequencies greater than 10° per second, and it is shown that under normal laboratory conditions a beat note can be obtained perfectly steady for large periods. Some applications of the method to physical measurements. are then considered, the changes in frequency being conse- quent on changes in. the inductance of one of the circuits. These latter changes were brought about by inserting a specimen within the coil, and were of two kinds :— (i.) An eddy current effect, the nature of which was examined for cylindrical specimens of different materials. (ii.) In the case of magnetic substances, a susceptibility effect. This latter variation was employed to measure the magnetic susceptibility of several salts at frequencies ranging * Théodoridés, Joe. cit. On Elastic Equilibrium under Tractions. 501 from 3 x 10° per second to 4 x 10° per second, and the results obtained are in good agreement with those previously obtained by static methods. My best thanks are due to Sir Ernest Rutherford for many suggestions connected with the problems and for his helpful criticism and encouragement during their progress. Cavendish Laboratory, Cambridge. May 20, 1922. XLVI. On the Conditions for Elastic Hquilibrium under Surface Tractions in a Uniformly Eolotropic Body. By R. F. Gwytuer, 1/.A.* ie a paper “On an Analytical Discrimination of Hlastic Stresses in an Isotropic Body,’+ I have expressed the elements of mechanical stress under tractions, and also the elements of strain, in terms of quantities which resolve on transformation of orthogonal axes in the same manner as elements of stress resolve. In this paper I propose to adapt the same method to uniformly eolotropic bodies. Briefly, we shall have under tractions, = ma 0’d3 nhs O°ds Ox! P= a2 AE +2 RE n= 10"; Or er > ON ae oz 02 ie Ox 52 es, 07d es 0°¢, : 0x3 a Ox? Oy’ ieee Oy’ S= OK ee OP y x SO7N3 01) 1DENs on Ode | Ozrd2 ‘Or0y’ T= O°X2 ee 0x3 Ms 0’d2 as 0x1 ol Oyorm O02 Oz dy’ U= Bx Dye BK , De Oe Oy O02 Nn 0e > O07 07’ (1) where {,, do, $3, ¥1 X2 X3} resolve as elements of a stress resolve. * Communicated by the Author. + Phil. Mag. July 1922, p. 274. 502 Mr. R. F. Gwyther on Conditions for If {01, 05, 63, Wi, We, 3} form another set of quantities, acting also as elements of stress, we may take as the com- ponents of an arbitrary displacement _ 0h: | Os Shh, 7, Sa. Oe Oe: _ Ov; | 082 | Ov - 2 Oy? oer on tis, Mw ai ae aye and therefore for the elements or strain 3°, ahs os e= Pye 45 Oa dz + 02 OY 9 Me 0°02 oy o's a oy” ¥ Oy Oz + Bray ” a 3%; Oth Deyo Lae 02 y OY Oz ui 02 02 ; ie O' , Ob . 0702+ 42) Os Os eis ey’ v 02? ne Oy 02 i OL 02 i Oroy ” _ Os "bro Os 0°03 + A) Ovi fie 02? + 02 f Oy 0 € 02 02 OL0y —~ ay Obs Os 07h, Oy 07(9; + Ae) Sera ae tS Ouiee + 3202, @ (2) Before proceeding, I will recapitulate that if the co- ordinate axes are rotated about their own positions through the small angles ,, wy, w,, and if 0,, O., Os (being partial differential operators) give the consequent coefficients of ®,, @y, ®, in {é, fg, a, 6, cor nm { P,Q) Rae. Uy aen 0 . OO Bie a(S 5) | ee eet in the case of elements of strain, and fe) o,= 28($ - 2) ao, uv. 4 78 in the case of elements of stress. Elastic Equilibrium under Surface Tractions. 503 Both {, do, ds, Xl» X2> x3 and {6,, 0», Os, Whi, Wro, vrs | are to follow the latter type. The relations between these two sets in elastic equilibrium must be linear. I shall take the invariant function (2V), which gives the potential strain-function per unit of volume in the form, 2V = Aye? + 2A + 2X13 eg + 2r44ea + 2X15 eb + 216 €C + Nog J? + 2ro3 fg +2 fa +25 fb + 2ro¢fe + 33 J? + 2r34. ga + 2r3590 + 2r36.9¢ + Ay a?+2r0yzab +2rggac + Nsz5 b? + 2rs6 be + Nes in (3) Since V is to be an invariant function, that is, since Q2,V=0, O.V=0, 0;,V=0, we deduce that, acting on the coefficients, ig fe fo) fe fo fo) a= Drus( 35 — —_ 5) = (i3— Az) Sree means ley ae fe) fe) fe) + 4r 0s) + (Ag3— a2) + (Ase — Ave + 2A45 ro + (Ass — X25 — 2A4e r3 = 0, | (6) with two other identities, whose consequences we can infer by symmetry. Thus Ay) = Age =Agg=A (an invariant), Noa = Ags = Ais = Avs = Me= Avs = 0, Nya ae 256 = 25 + 246 = 36 ote 245 = 0. ° . (7) Also, as there is no term in fa or ga in 2V, the coefficients of 0/OA, and of 0/ds4 In Q, are separately zero, and A2g=A—2Ayu, Ats=A—2A55, Arg=A—2Abge. . (8) Elastic Hquilibrium under Surface Tractions. 505 We therefore find OV =X(o+f+9)? + ru(a?—4/y) + Aso(b?—4eg) + Ass(c?—4e/) + 2r56 (be— 2ae) + 246 (ac— 2b/) + 205(ab — 2cq) ° () and 0 fe) fe) fe) O = —2rso( aaa N55 — Age) 3. — — Au55 — } "\OAss — OA6s ( Mckee lis Bas fe) a © OAs I shall now write, for convenience, Agg=V1, A55=V2, Ase=V3, ASE= — V4, Ats= —V5, Ass= — V5 and 2V = rle+f+y)?+ r1(a—47y) +2(08—4eg) + 04(e—4ef) + 2v4(2ae —be) + 2v5(2bf—ae) + 2vg(2cg—ab), . (10) where es 0 fe 2 = 2n (3 a = 2) 4 (4- Pee Ome The requirement that the solid os act as an elastic solid under tractions has reduced the 21 eolotropic constants in (3) to 7 constants of elastic equilibrium in (10). Accordingly, we find frum (5), oi = V\(O9+ 03) —vstro— Vows, fz = Vo( B24 41) —vyiri — ves, d= v3(0,; +05) —vyhr — Vso. . . : (TE) The value of 2y, is deduced from 0,¢,= 2, = v4( 20; + 02+ 83) —(ve+ v3) Wi —VeWo—vs3, with 2X = V5(41 + 202+ O3)— veri — (11 + V2) Wo— aps, 2y3 = ve( 0; 9 0, + 203) —v5y i Varo — (v, i Vo) Pr. ( 12) From the value of 2V given by (10), we tind the elements of elastic stress, as usual, by differentiation, P = re+ (A— 203) f+ (A— 22) 9+ 2y,a, Q = (A—2vz)e+ Af + (A—2)g + 2050, = (A— 2y.)e+ (A— 21 f+rAG + 2vVec, S = 2vye+vja—v,b—vsc, T= 2v3f—vea t+ vob —v4c, U=2y,.9—v,a—vwb+vsc. .*. . . . ~ (18) 306 Mor. i. 3 Gwyther on Conditions for To complete the equations, we substitute for {e, f,g, a, 5, c} from (2), equate to the mechanical elements of stress given by ( 1), and substitute for pis bo, Pe X1> X23} to find the S1x equations for condition in {6; 2, 03, Wi, Wo, 3}. This general process will be now simplified. In order to carry out the scheme of this paper it has been necessary that the axes should be arbitrary and the stress- elements should be written in full. But this stage having been completed we may now select a special set of axes, and also simplify the arbitrary stress-elements. ee {e, f, g, 4, b, c} act on transformation of axes as ue Up ry AOS Qe, Qayt act, as do also {a’—4fg, b?—4eg, —Aef, Awe —2be, Ab f — Zac, 4cq—2ab}, and since a CEOs io 3 3 o— Dia ty sate) Serene ays + Vaae we find that V2? + Voy? + v3e" + 2vgyz+ 2vsve + 2veny = 1 - is an invariantal ellipsoid, and the principal planes of this ellipsoid are planes of elastic symmetry in the body. I shall take these principal planes as the coordinate planes, and consequently v,=0, v;5=0, v,=0, while 1, v2, v3 now stand for the roots of the Discriminating Cubic. The equations (11), (12), and (13), become for these axes, do, = V1( A, +E 03), 2M) = = (V9 v3)Wi1, do = v2( 03+ 0), 25 = — (¥3+11)ho, d3 = v¥3(0,+ 85), 23 = — (1. + 2)s, P = re+ (A — 203) f+ (A— 202) 9, Q = (A—2vz)e+rAf+(A—2n;)g, = (A— 2, )e+ (A— 2”) f+ AQ, S = vas TS 95b, WW ac ee It will now suffice to put ~,=0, ~.=0, 3=0, so that .. OOF “OU: _ 063 3) ee eae tS) Ue p) Den ee OZ and it follows, from above, that y,;=0, ¥.=0, y3=0 at the same time. We shall therefore remain with only three equations in 0,, 0, 03, when these simplifications are made. Elastic Equilibrium under Surface Tractions. DOT These three equations take the form B81, B,D) __, BULB) _ B%G—8 Se + OT 32 )= hoa a 3 =, BUG =61) _, 9°(0.— 8) ay 3 Ou? 022 ? _ BOO) _ 3° —4,) Vo Bu? 1 Oy? ’ so that ety ct era HELD.) v1V7?(@,—03) = 2 (6, — 43) + v,(A3—0,) +v3(0;—O.) }, with two similar ee and 4.02 .v20y" : =) fy1(8—03) +¥9(85—0,) erg TS LUN AUS Hence v,(@.—63) + v2(@2— 01) + ¥2(@,—92) is an Ellipsoidal Harmonic, from which 6,—6, 0,— 63 can be deduced. In a previous paper, I have dealt with the simpler case for an isotropic body, when vy; =y,=v3=n and 6,—63, 6.—93 are Spherical Harmonics. The present equations apply to crystals having three orthogonal planes of elastic symmetry. The investigation seems interesting because it passes some- what outside the range of elastic equilibrium, even if it is ultimately confined within that range. If we regard a piece of sound material intended to serve as a test-piece we cannot consider its potential energy of strain to be precisely in the form given by {10}, although by judicious working it tends to approach thatform. The actual potential energy has pro- bably a form such as that in (3), until it has been worked. The effect of “working” simplifies the form of V in a manner perhaps comparable with the algebraic discarding of constants inconsistent with elastic equilibrium. If this comparison is not unreasonable, we may venture to extend the idea, aud to imagine that an excessive exertion of trac- tion may again alter the form of the potential energy, and introduce into V terms inconsistent with elastic equilibrium and may, if continued, lead to rupture. At any rate, the theory of rupture must lie outside the range of elastic equi- librium, though not necessarily outside the elastic stress- strain relations. | 908 1 XLVI. On the Viscosity and Molecular Dimensions of Sul- phur Diomde.” By C.J. Suive, B.Sc; ALkICsS, aes Research Student, Imperial College of Science and Tech- nology, London ”*. ECENT work on the viscous properties of compounds which are ordinarily gaseous having been successful in elucidating the molecular structure of these compounds, it was thought that it would be interesting to apply similar methods in the case of sulphur dioxide, especially as. Lang- muir f has already suggested a possible arrangement of the atoms which constitute this particular molecule. This paper describes the necessary viscosity measurements for sulphur dioxide. Previously the data regarding the viscosity of this vas were very scanty and did not extend over a sufficient range of temperature to determine Sutherland’s constant—a’ factor of almost as great an importance as that of the viscosity itself in determining the mean collision area of a molecule. Apparatus and Method of Observation. The apparatus and method of observation which have been used for the purpose of measuring the viscosity of sulphur dioxide have recently been fully described {. Method of Experiment. The mercury pellet, which was used to drive the gas through a capillary tube which forms part of a complete circuit con- sisting of this tube and a fall tube in which the pellet moves between specified marks, is the same as that which was used by the author in his experiments on carbon oxysulphide §. The time of fall for air proved to be 105°53 secs. at 18°0° C. With this time of fall the corresponding time of fall for sulphur dioxide has been compared, and, with appropriate corrections, this gives the relative viscosity of air and sulphur dioxide. From this relative value the absolute viscosity has been obtained by assuming that the viscosity of air at 18°:0 ©. is 1°814 x 1074C.G.8. units. In addition, the varia- tion of viscosity with temperature has been derived from comparisons of the corrected times of fall at atmospheric and steam temperatures. * Communicated by Prof. Rankine. + Langmuir, Journ. Amer. Chem. Soe. vol. xli. p. 868 (1919). t A. O. Rankine and C. J. Smith, Phil. Mag. vol. xli. p. 601 (1921) ; and C. J. Smith, Proc. Phys, Soc. vol. xxxiv. p. 155, June 1922. § C.J. Smith, Phil. Mag. vol. xliv. p. 289 (1922). . See eee eS eo > re — Molecular Dimensions of Sulphur Lnowide. 509 Preparation and Purification of the Sulphur Diowide. The sulphur dioxide was generated by the action of dilute sulphuric acid on sodium sulphite. It was dried by being passed through several wash bottles containing concentrated sulphuric acid, and then solidified in a U-tube surrounded by liquid air. The U-tube was then cut off from the generating apparatus, and all permanent gases removed by means of a pump. The gas was made to evaporate by removing the liquid air, and samples were collected over mercury. The. chief difficulty in using this gas is to dry it sufficiently well that the motion of the pellet of mercury in the fall tube shall be smooth. It is difficult because sulphur dioxide boils at —10°C., and at this temperature water has a small vapour- pressure. It was finally purified and dried in the following way, before introduction into the viscometer :—The gas was solidified in a tube maintained at —80°C. by means of a mixture of solid carbon dioxide and alcohol, and all per- manent gases and possible traces of carbon dioxide removed by means of a Toepler pump. Sufticient alcohol was then added to the carbon dioxide mixture to raise the temperature to —60°C. At this temperature water has a negligible vapour-pressure, while sulphur dioxide is liquid and has an appreciable vapour-pressure. This enabled successive small quantities of the dry gas to be pumped off and introduced into the viscometer, which was previously exhausted, until the pressure therein was atmospheric. Heperimental Results. TABLE I. Each time recorded in this table is the mean of four observations in each direction for the whole pellet, and of three when the pellet is divided into two segments. The letters in parentheses indicate the order in which observations. were made. | Time of fall (secs. }.. ‘Time at | ‘Temp. Capillary | Corrected | (deg, C.). | Whole | Two | correction. time | 18°:0C.|100°-0C. | pellet. |segments.| (a). (7). fe ee eee eS a EC (a) trie se16 | 79°67 | 00422 72:95 73:03 |(6) 17°80 | 7648 | 80°38 | 00463 72°95 73°01 | (ce). 17°35.) 76°82 81:00 | 0:0491 73°05 73°09 (Ff) 17°38 | 7640 | 80°37 | 00471 72°80 72:97 | | Mean|...73°03 1(¢@)1000 | 97:10 | 99:32 | 0:0219 94:97 ai, 94°97 (e) 99°96 | 97°13 99°37 |. 0°0220 94°99 Sey 1} ye eee | Mean ...94°99 o10 Molecular Dimensions of Sulphur Dioxide. We have t)3= 73°03 sec., and t)9)= 94°99 sec. The ratio of the viscosities at 18°-0 C. and 100°:0C. is given by the ratio of these times ; thus Mion 0p ee ms tig 13°03 Assuming Sutherland’s law to hold over the range of temperature used in these experiments, the value of Suther- Jand’s constant is 416. The validity of Sutherland’s law for this gas over the range of temperature investigated cannot be expected to be great on account of the probable large deviations from Boyle’s law which this gas may exhibit, since the temperatures at which measurements have been made are not very far removed from the boiling-point of liquid sulphur dioxide. The value of C given, and subsequent deductions depending thereon, should therefore be accepted with some reserve. al) le oe ts0. __ 73°03 _0:f09 Also at 18°70 C., i = 10553 =() 6923, Correcting for slip in the usual way, we obtain Nair Assuming that the viscosity of air at 18°0C. is 1°814~x 1074 C.G.S. units, the values for SO, are Me 1°253 x 107-4 C.G.S. units, Mion = L630 x 10%" CxG. 5. anits, and by extrapolation, using Sutherland’s law, no lose L0e. CaG.Sammits, According to the usual works of reference and published papers, Vogel * is the only modern worker on this subject, and he found that m)=1:183 x 1074 C.G.S. units. Calculation of Molecular Dimensions. The above results enable us to calculate for sulphur dioxide that mean area which is interpreted by Professor Rankine Tf as the area which the molecule presents in mutual collision with others. Chapman’st formula, modified in its interpreta- ction, as indicated above, is ‘the basis of this calculation. The * H. Vogel, Berlin Diss. p. 46, 1914. + Proc, Faraday Soe. vol. xvii. part 3 (1922). { Chapman, Phil. Trans. A. vol. cexvi. p. 347. Simple Model to lllustrate Elastic Hysteresis. oll value obtained is A=0°94 x 10~” cm.?, which may be subject to an experimental error of 2 or 3 per cent. It is difficult to estimate the degree of precision with which this figure represents the real dimensions of the molecule. The mea- surements of viscosity, owing to the comparatively small temperature range over which they extend, provide no. proof that, for this gas, Sutherland’s law holds. Indeed, as men- tioned earlier, it is improbable that the sulphur dioxide in the cireumstances of the experiments was sufficiently super- heated to give the true value of Sutherland’s constant. It is not unlikely that the actual mean collision area differs from that calculated by an amount appreciably greater than that attributable to experimental error. Summary of Results. TABLE TI. | Viscosity in C.G.S. units x 1074, Mean col- | Sutherland’s | lision area 18°00. 100°°0 C. 0°-0 0. constant. |(em.?x 10-15). 1:253 1°630 1168 . 416 | 0-94 In conclusion, the author would like to record his apprecia- tion of the continued help and advice received from Professor Rankine, and also to thank the Goverment Grant Committee of the Royal Society for a grant which enabled the research to be undertaken. Imperial College of Science and Technology, London, S.W.7. 10th June, 1922. XLVI. On a Simple Model to Illustrate Elastic Hysteresis. By 8. Luss, M.A., St. John’s College, Cambridge *. ; § 1. Introduction. \ | UCH material has accumulated + in recent years con- cerning the behaviour of metals when taken through either a series of cycles of alternate compressions and tensions, or a series of periodic shear stresses. In the main, the experimental results here utilized are those of * Communicated by the Author. T See e. g. ‘ Dictionary of Applied Physics,’ yol. i. p. 178. 5 Mr. 8. Lees on a Simple Model Messrs. Smith & Wedgwood, ‘Journal of Iron and Steel Institute,’ vol. xci. p. 374. It must be admitted that in one important respect, the results of these authors appear to differ from those of other experimenters; in that, according to the cited authors, elastic hysteresis would appear only to make itself evident under certain conditions, whereas other experimenters have been of opinion that elastic hysteresis * always occurs with stress change, even with small range of - stress. Without expressing any opinion on the existence or not of elastic hysteresis with small ranges of stress, the author has taken the results of Smith & Wedgwood, and attempted to construct a simple model illustrating these results. Whilst it is not contended that the model to be described is the best possible, it does to some extent satisfy a desire to reduce to simple mathematical treatment many of the well-known elastic phenomena. It may be here noted that the model described below in § 3, and the modification of § 9, between them illustrate a whole series of well-known phenomena, such as: (j.) Existence of elastic hysteresis loops only under certain conditions. 3 (ii.) Small variation of area of loop with speed of description. (iii.) General shape of loop. (iv.) General character of mean-stress-strain loops obtained when slow speed periodic stress is combined with rapidly alternating stress. (v.) Existence of two points in loop at which (limits of Hooke’s law) the loop commences to depart from the straight line law. (vi.) The production of permanent set in a material and the existence of a true elastic limit beyond which elastic recovery 1s infpossible. (vii.) The existence of a yield point. (vii.) The general relationship between elastic hysteresis and the three conditions ‘referred to in (v.), (vi.), and (vil.)). | (ix.) The phenomena of slip bands. (x.) The effect of overstrain on the two points of (v.). So far as the author knows, such a model with its illustra- tions as above, is original. Before Aeecoib in the model, some reasons for ruling out ordinary viscous fluid effects as the predominating cause of elastic hysteresis will be given. * Diminishing, of course, with the range of stress. to Lllustrate. Klastic Hysteresis. 513 »§ 2. Discussion of the Problem. In connexion with elastic hysteresis, it seems natural. to invoke viscous effects analogous to those found in viscous fluids. A very obvious idea. is to introduce fora material undergoing cyclical variations of stress, a stress term always depending upon the rate of change of the corresponding strain. Thus, e.g., if f denote stress, s denote the corre- sponding strain, we may try ji Ks As) Wace (1) where K and XY are constants. For the type of stress con- sidered, K would be the ordinary modulus of elasticity. If we now make s go through a cycle given by (¢=time) Sr ESHCOS Doth tad ati sis. CD) we get different values of f which can be plotted against s, giving rise to a stress-strain loop*. This loop is clearly an Fig. 1. ellipse, asin fig. 1. For on eliminating ¢ between f= s(K cos pi—Apsin pi) ... - . (8) and (2), an ellipse arises. It is not difficult to verify that the area of the ellipse is proportional to both p and s). Thus with such an assumption, the area of the loop, for a given 8, will diminish indefinitely as the speed of fluctuation * In plotting stress-strain loops, we can always make the straight line f= Ks have any slope we please by suitably choosing the scale for fands; but in general this will give us a loop of minute proportions, for breadth. Having chosen a suitable slope for the above line, the loop can be magnified by representing the divergence ( f—Ks), of a point of the loop from the straight line corresponding to no hysteresis, on a scale any number of times that of f. This will be assumed to have been done for the stress-strain diagrams shown in this paper. Phil. Mag. 8.°6. Vol. 44. No. 261. Sept. 1922. 2 L 514 Mr. 8. Lees on a Simple Model is caused to diminish indefinitely. But this is quite contrary to the facts obtained by experiment. The results obtained by using a formula * of the type f= Kets + pets + cc, et ae eee are quite analogous. On putting s=s)cos pt as before, we shall get Sf = 5 {cos pt (K—pp?+ ....)—psin pt (A—vp?+ ....)3, (5) and on plotting f against s, we shall again get an ellipse. Further, whilst the area of the ellipse will not follow quite so simple a law as before, it will easily be seen that as p is made to approach zero as limit, the area (for a given 59) will do likewise. This, again, is contrary to experience. It may be said, in passing, that elastic hysteresis loops are not found in exact experiment to be ellipses at all. A nearer approximation to actual shapes can be found by assuming the loops to be lenticular. Such a shape can be got, e.g. by taking fH=Kst BF, 003. Co where the sign of the 8? term is always so to be taken as to. make the frictional stress term oppose the change in strain. ic y2 Taking s = s) cos pt as before, it will be found that we get a diagram which is the result of eliminating ¢ between this relation and f = Ks cos pi bsp" sin? pts.) 2) 1 If the loop be considered to be described in the clockwise sense, the plus sign will be taken from t=—7/2p tot=a/2p. The loop is then seen to be lenticular, as in fig. 2. Here, again, * See Maxwell, ‘Collected Papers,’ vol. ii. p. 628. to Illustrate Elastic Hysteresis. o15 the area of the loop vanishes with p, and so violates the facts. A combination of formule (4) and (6) will suffer a like disability. In view of the undoubted fact that when the strain range is large enough, hysteresis exists, whether the speed of describing the “eycle of strain change be quick or slow, the author has felt it necessary to fall back on the rather primi- tive notions of solid friction. It is not contended that the ordinary ideas of fluid friction do not enter into the produc- tion of elastic hysteresis at high speeds, but it is asserted » that with low speeds of describing a cyclical Guana? fluid friction effects are negligible. The author has found it possible, using the notions of solid friction, to get results in good qualitative agreement with many of the facts, and these notions Spelt mathematical analysis of a simple character to be applied. The chief factor in solid friction that is made use of in this paper is the property according to which there is a marked discon- tinuity between the limiting tangential force just required to produce sliding (of one surface over another pressed against it) in one direction as compared with the opposite direction. If the tangential force available lies between these limits, no slipping takes place. In fluid friction, as usually understood, the frictional force will vary continuously as the direction of relative motion or sliding is altered, and at the instant at which the two surfaces have no relative motion, the tangential force exerted by one on the other will momentarily vanish. This behaviour is totally different to that obtained with the agency of solid friction. § 3. Description of a Simple Type of Model. In the model now to be described, the author conceives that in a metal under stress there are groups of molecules (or possibly crystals) which in some way are capable of recelving and transmitting directly a portion of the stress applied. There are also other groups of molecules which may take up a portion of the stress, but this portion is deter- mined by considerations of solid friction. The solid friction is supposed to arise owing to the pressure of the first set of groups acting normally on the second set. ‘This pressure may be considered as molecular in origin, and of the same character as cohesive effects. Such pressures will doubtless be large, and as a first approximation we may suppose that this pressure is not materially altered even if slipping takes place between members of the two sets of groups. It may 212 516 Mr. 8. Lees on a Semple Model further tentatively be assumed that whether or not slipping takes place, the two sets of groups will behave elastically, so far as each set is concerned. These points are involved in the model shown in fig. 3. In this diagram, A, and A, represent two groups of molecules of the first kind referred to. A,and A, are directly connected up to the agency pro- ducing the stress (represented in the diagram by forces I’, F), and are shown joined together elastically by a spring (marked with tension T, in the diagram). As typical of the groups of the second kind referred to, we have the items B,, B,, Cy, and C,. Here B, and C, are between them squeezing A, with a pressure N. B, and C, are performing Fig. 3. Springs to produce compressio7 VV a like office for A,. B, and B, are shown joined together elastically by a spring (represented as being in tension T,). C, and C, are similarly connected together by a spring (also shown in tension T,). The diagram is intended to represent a state of affairs such that when F is zero, the tensions T, and the tension T, all vanish. Such a state of affairs may be called the neutral state. For such a state, the dis- placements of A, B,, and C, from their equilibrium positions are all equal, and (measured with OO, taken as unaffected) may be denoted by w. Similar remarks apply to A», Bg, and C,. Since we shall have under these circumstances MN, = AL, MM = Now, e ° . e e (8) where A, and A, are appropriate elastic constants, it follows that = (A, at 2X2) a. siueiie tal tte let hagite ° (9) to Lilustrate Elastic Hysteresis. 517 Thus regarding F as representing externally applied stress, and w as the corresponding strain, we get the usual Hooke’s law holding for stresses which do not disturb the neutral state. This remark, of course, holds for either tension or compression. If w denote the coefficient of solid friction between A, and both B, and C,, the same value holding for A, and both B, and Cp, slipping will take place when a OLENA Lat solr iweaea” ab ae vO) This will correspond to a value of a given by pens Fis NG Thus the value of F at which the linear proportional relation- ship between F and z breaks down will be 2 F = 0,42a4)2 = Cot 2h) py, ah (Gay 2 if F be increased beyond this limit, slipping will take place, of amount (say) y, between A, and B, and C,; also (see fig. 3a) between A, and B, and O,. The tensions T, will remain constant during such an increase of F, and we shall now have | F = Ayw+ 2uN = Aye + 2r(av—y). ° ° (12) This, though a linear relationship between F and #, has a different slope from (9), and F is no longer proportional to w. 318 Mr. S. Lees on a Simple Model § 4. Hlastic Hysteresis Loop for Model. After slipping has occurred as just explained, the moment F is caused to diminish, slipping will cease, and A,, B,, and Q, (also A,, B,, and C.) will move together. Thus if Fj, 2, be the values of F and 2 at the instant that F is caused to decrease, we shall have at first F,—F =, 42))\Geyee)) (ee During any change of the type given by (13), there will be a constant amount of slip (reckoned from the neutral state) given by ie) 2 ee YN a ons oe Uy SOLS ° e (14) It will be noticed that during the change given by (13), the slope of the I’, # curve is exactly the same as that given by (9), 2. e. as that for the neutral condition, also T, is now at any instant given by T,—pN = Ao 4% — 21), ° . ; ° (15) since the slip does not alter, and dT./dx = 2g. | As F goes on diminishing, ultimately becoming a force of compression, a time will arrive when slipping will once more occur. This will clearly be the case when To —yN, 2. or from (15), when p=— (7 —«,). al laPa a at Chi} Ae This corresponds to a value of F' given by F = ye 2uN 2a (= +1) Mee 2 This result may be compared with (11). It will be noticed that the value of F given by (18) is not in general the nega- tive of that given by (11). It is easy to see that further compression beyond the value given by (16) will result in a straight line law for F and a, such that ee) Make aw 4 a its) he For such further compressions, of course, sliding takes place. If now the compression be gradually diminished, 2. e. F be to Illustrate Elastie Hysteresis. 519 increased algebraically, we shall again go through a series of operations in which first dk : ae Neanane ane (aieisu by (20) after which (17) will hold. By performing such reversals of stress, between limits +F,, sufficient to cause slipping to occur in both directions, we can ultimately reach a cyclical condition; the stress- strain loop, 2. e. the F—w loop, being as shown in fig. 4. Fig. 4, - Xp7F) The diagram is such that the limits of F are +F), the corresponding limits for # being +2}. The typical feature of the cyclic condition is, of course, that the F—.z loop is symmetrical with respect to the origin. In our case, the diagram is a _ parallelogram JKLM, with dF/dx=X, for the lines KL and MJ, whilst d F/d«=(d,+2A,) for the lines JK and LM. To get the F—-2 coordinates of K, the intersection of the lines JK and KL, we have for the line JK 3 F—(—F)) = (A, + 2A) [a—(—2,) }, ae (21) whilst for the line KL we have FL—-F = Ay4(a,—2). ee an) aM Baer (22) These two equations give for the point K :— ESI 220 00 ( rg ee pts (24) « \ | j 520. Mr. 8. Lees on a Simple Model For the point M, the values of x and F are clearly the eee tives of (23) and (24) respectively. The value of the maximum slip (measured from ‘the neutral state) can easily be obtained, for this must vary between (say) y; and —y,. The range of slip is accordingly 2y,, and this range of slip is incurred during the description of both the lines KL and MJ. Thus 2y, must equal the difference between the values of # at Land K, ?.e. Bed us Az)@ A, +2As)e—Fy (25) 2 Ae This result can also be obtained from equations (12). Yet another expression can be obtained by remembering that at the point L, where the slip is ¥;, we must have from (12) 2y; = 14 — Ao(a1— 71) = WN. Substituting the value of y given by (25), we get * Fy— 2A, = 2yN. ue (26) § 5. Area of Hysteresis Loop. The area of the hysteresis loop can now be obtained. Itis clearly twice the area of the triangle JKL, the coordinates (2, F) of the vertices of which are respectively (—2a#, —F)), Ce NE, aa ae Se No Ae and (a, Fy). The area + is therefore | ae ee Fy (Ay # Ae) 21 FiOg +2) —Mi(Ay + 2Ap) ay s Bik hii tu ler Voaumaa. GT Ag re zy ) 7 F, ah | {tall 2(F,—ayn gg ws ane Doig cy Fy] = 8¢#Ny1, + (27) on using (25) and (26). - This result might also have bese obtained by remarking that the work lost in a complete cycle can be accounted for as due to a force of friction 24N overcome twice (during * When hysteresis exists, the locus of points L is therefore a straight line. + The negative sign is used so as to make the area positive. to Illustrate Elastic Hysteresis. 521 the stages KL and MJ) through a slip on each occasion of amount 27. The area of the hysteresis loop clearly vanishes when y,= 0,7. e. when F, = Oy a 2). . . . e ° (28) Using (26), we get for the corresponding limiting values of &y and F, —— pN XY = ies " 1 29) Fi= NOY +- 2r2) 1 ; } ? 5 Xe The limiting loop for which the area just vanishes is shown in fig. 4 as the straight line PON. It is easy to verify from. (23) and (24) that N is the middle point of KL, and similarly that P is the middle point of JM *. It is to be understood that for values of 2, aad F, less than those given by (29), the F'—w relationship is always ex- pressed by a portion of the straight line NP, whose slope is the same as that of both JK and LM, and is that cor- responding to the neutral state. This statement arises from the fact that for such values of # and F, there will be no sli F ae (26) and (27) we can get two oe expressions for the area of the hysteresis loop (when it exists) in the forms: N\ _ 8uN N(Qro+2 uN (x.— BA) = KS [pe ). . (30) From these we see that the area has a value proportional] to the excess of 2, over the limiting value just referred to; alternatively, it is proportional to the excess of He over. its corresponding limiting value. § 6. Comparison of Results obtained with Experiment. A large amount of data is available concerning elastic hysteresis, but, for the moment, reference will only be made to the paper by Smith & Wedgwood, loc. cit. figs. 2-5. The static stress-strain loops obtained by these authors for tension and compression of a material in the cyclic state agree in several particulars with the theory just outlined. Thus elastic hysteresis does not occur till the limits of stress * To get the model to the neutral state from the cyclical state, we arrange for it te be put into either of the conditions corresponding to the points N or P, and then take off the load F. 522 Mr. 8. Lees on a Simple Model exceed numerically a certain amount, and when this does occur, the area is proportional, for small-sized loops, to the excess of I, over the critical amount. Also, immediately after a change in sign of di'/dt, where F represents stress and ¢ the time, the stress-strain curve is always straight and parallel to the ‘strai ght line which represents. elastic change without hysteresis. On the other hand, the author’s theory fails to explain two noticeable points indicated in Messrs. Smith & Wedgwood’s aper :— . (1) When a point corresponding to K in fig. 4 has been reached, the stress-strain curve is actually found to become Fig. 5. curved, as shown in fig. 5, instead of following a straight line like KL of fig. 4. (2) For large areas of loop, it is actually found that the proportionality between area of loop and excess of F over the critical amount above referred to breaks down. Other divergences between experiment and the theory at present outlined will be indicated below (§ 9). § 7. Steady Hysteresis Loop for Unsymmetrical Stress Limits. We shall now discuss the nature of the steady hysteresis loop when the limits of stress are not equal and opposite. The appropriate stress-strain (in our case, '—2) diagram 1s indicated in figs.6 and 6a. From the considerations out- lined in § 4, it will readily be seen that just previous to F reaching its lowest value (as at J), the F—vw relation will be (on the theory outlined) dF/da=X. The corresponding line to Illustrate Elastic Hysteresis. 523 in the diagram is MJ. . After F has begun to increase, the F—zx relationship will correspond to the straight line J, K,, whose slope is given by dF /dzv=2,+ 22. It will readily be seen that the steady loop ultimately described will be a parallelogram like J,K,L,M, of figs. 6 or 6a, the .slopes Fig. 6. of whose sides have been indicated, but the corners of which have to be found. Referring to the model shown in fig. 3, it will be understood that in describing the loop, the slip y will change in one direction and the other alter- nately. When slip is occurring with increase of F, the 524 Mr. 8. Lees on a Simple Model H—z relation will always be given by a portion of the line KL of fig. 4. Similarly when slip occurs with diminution of F, the F—z relation is given by a portion of the line MJ. The actual loop for unsymmetrical stress limits will therefore be obtained by choosing the points L, and K, on these lines respectively, so that the stress at L, is Ei+F, and at J, 1s F)— F,, where F)+ F, are the unsymmetrical stress limits. It will readily be perceived from the diagram that the area of the loop will only depend ‘on the range of variation of stress, 7.e. on F; ; and fora given value of I, the diagram is got by taking F,=0 (i.e. for the cyclic condition), and displacing it parallel to the line KL of fig. 4 until the stress limits come right. In particular, the results of §§ 4 and 5 for the area of the loop in terms of F will hold, provided we interpret IF’, as being one-half the greatest variation in stress, We may conveniently term the state of affairs here referred to the asymmetric cyclic state. For such a state we may briefly indicate the results involved. _ From equation (29), provided our assumptions hold, we see that for no hysteresis to occur, }' must always lie between FEN Qu hs) ee 2 where, of course, F, may have any fixed value. When this condition (31) is not satisfied, the area of the loop is given by (30), where F, has the meaning of half the greatest variation of stress. § 8. Particular Case of Fo= Fj. If the lower limit of stress for the asymmetric cyclic con- dition be zero, we get an interesting case. We have here to take Fy>=F,, and hence Fyay,=2F,. From (31) we see that the greatest value of F for no hysteresis to occur will accordingly be 2MN(Ay oP 2r2)/Ao- o Misre VT (32) This is exactly double the maximum stress for the cyclic condition, which just fails to produce hysteresis. Tf the stress varies between O and F, and hysteresis does occur, equation (30) shows that the area of the steady loop will be given by AuwN 2uN(2r,.+Az) <|F- i. i eS It is thus proportional to the excess of F over the value (32). to Illustrate Elastic Hysteresis. D925 §9. Effect of Rapid Periodic Changes of Stress combined with Slow Variation of Mean Stress. If in $7, we imagine that F) changes slowly and _ periodi- cally whilst the F—- loop is being rapidly described under the influence of a constant Fy, we get an important case in experimental work. In such a case, we may imagine that the variation in Fy is so slow and the periodic change of superimposed stress (between limits +f) so rapid, that at any instant the material is always in the asymmetric cyclic state. It is interesting to discuss the loop obtained (if any), by plotting the different values of I, against the corre- sponding mean values of «x for the rapidly described hysteresis loops. We may call such a diagram a mean- stress-strain loop (see Smith & Wedgwood, loc. cit. p. 318). There are three cases to consider :— (a) If the limits of stress F)+F,, Fo—Fy, are such that the corresponding points in the F—wz diagram lie between the lines KL and MJ of fig. 4, then the mean-stress-strain Fi a. 7. F L loop reduces to astraight line. This straight line will either be a portion of the line PON, or a portion of some line parallel thereto. For the neutral condition, the points P, and N, which limit this type of mean-stress-strain curve, will clearly have values of I) given by minus and plus respectively an amount given by subtracting the amplitude of F, from the value of F at N (see fig. 7). (b) Using | F,| and | F,| as amplitudes or positive limiting values, we may still consider the case of |F,! less than the 526 - Mr. §. Lees on a Simple Model value of F corresponding to the point N ; call this value Fy (given by equation (11)). Keeping |F,| constant and slowly allowing F, to increase from zero, we ultimately get (on the mean-stress-strain curve) to the point Nj, corresponding to the maximum stress Fy. Further increase of the maximum stress can only be obtained with slip, and such maximum stresses must correspond to points lying on the line KL. During the rapid variation of F,, whenever the F—« point on the diagram leaves the line KL, it will do so to travel along a line (like RS) parallel to PON, and must return to the line KL at the same point that it left it (since during this travel there is no slip). Hence the line of the mean-stress- strain curve corresponding to the description of the line KL will be Kgl, a line parallel to KL (see again fig. 7). The complete mean-stress-strain diagram for the complete cyclic variation of I, is therefore given by J,K,L,M,. It will be - noticed that it is of similar type to the static stress-strain loop of fig. 4, 2. e. its sides are parallel to the sides of the parallelogram JKLM of fig. 4. If we draw in the static hysteresis loop J;K3;L3M; cor- responding to the range +(|F,| + |Fi|), we see at once that the area of the mean-stress-strain loop J,K,L,M, is (Fy—|F,|)/ Fn times that of J;K,L;M;3, and is therefore by (380) given by . Cee | (Fol+1 Fil) — aos |: (34) (c) We lastly consider the case of | F,| greater than Fy. In this case, the rapid variations in | F,| will always cause the F—z2 or state point of the model to move from the line KL to the line MJ (or vice versa). A point on the mean- stress-strain curve is therefore always to be regarded as the geometrical centre (or centre of gravity) of a loop in the form of a parallelogram like J'K’L'M’ of fig. 7a, such loop corresponding to some asymmetric cyclic condition. The locus of such centre of area is clearly a straight line UOV, passing through O and lying parallel to KL or JM. In such a case, therefore, the area of the mean-stress-strain loop vanishes. For a given |Fo| and |F,|, it is easily verified that the x coordinate in the diagram (fig. 7a) of the end point V of this line is given by | F,|/2,, the corre- sponding value of F being | F9|. We may compare these results for mean-stress-strain loops . with those given by Smith & Wedgwood (loe. cit.). to Illustrate Elastic Hysteresis. 527 The two sets of results are in many ways in agreement. Smith & Wedgwood found that, provided the amplitude of F, was not too great, the mean-stress-strain curve was a straight Fig. 7 a. line for values of | F,| which did not exceed certain limits depending on |F,|. These limits were sharply defined, and when |F,,| was caused to go outside these limits, a mean-stress-strain loop was formed. A typical diagram is shown in fig. 76. Such a loop, which increased in size with 528 Mr. 8. Lees on a Simple Model inerease of | Fo], was generally quite similar-in character to the static stress-strain loops referred to in § 6. , The limits. just referred to correspond to P, and N, , of our fig. 7. If we take the static elastic limits for zero hysteresis loops given by Smith & Wedgwood’s fig. 14 as 9°5 tons per in.’, and consider the case indicated in the same authors’ fig. 15, wiz. | #;| = 7°65, we ought by (a) of this section to get for the point N, of our fig. 7 a value of 0-0) — oo) toms Mer dina: The actual value indicated in Smith & Wedgwood’s fig. 15 would appear to be about 1°6 tons per in.’. A difficulty arises in connexion with the differences in the stress limits for zero hysteresis loops as shown in Smith & Wedgwood’s figs. 2 and 14. The two results of these figures can be to some extent reconciled by assuming that repeated slip causes some change in the value of mw in the model, e.g. by temperature change, etc. Ultimately, of course, the explanation is molecular, but the fact that rest causes the material to go from a state corresponding to Smith & Wedgwood’ s fig. 2 toa state corresponding to their fig. 14, is a good reason for taking this simple explanation. There are certain outstanding differences between the results obtained from the model of § 3, and the actual results of Smith & Wedgwood. The first is that the mean-stress- strain loop is not a parallelogram in actual experiment. This has been already noted. Further, when the loop does not reduce to a straight line, it is found experimentally that for a given | F,|, variation of | F)| causes the points of maxi- mum F', (corresponding also to points of maximum strain) to lie on a curve like N’‘L’ of fig. 7b. According to the present theory, this curve should be a straight line (parallel to KL of fig. 4). In an attempt to meet the objections just raised to the model of § 3, and those mentioned at the end of § 6, the author now puts forward a slightly modified theory and model. § 10. Modified Model. Fig. 8:shows a modification of the model described in § 3. It will be noticed that the contacts for A, with C, and B, are point (or line) contacts ; similarly for A, with C, and B,. The rubbing surfaces of A, and A, are now taken as curved, , instead of straight (or plane). to Illustrate Elastic Hysteresis. 529 Fig. 8 shows the model in a configuration corresponding to the neutral condition, whilst fig. 9 shows a state of eo, S, affairs when slip y has taken place. Fig. 10 gives an enlarged view of one of the two rubbing surfaces of A»v>N uw). The real positive value of y can also be shown to exist by putting successively y=0 and y= in the left-hand side of (42), when there appears a change of sign. Bie As y goes on increasing, a time will come when T, will vanish. This, from (41), will clearly be the case when y=p/2a, and P= wp/2a ... . (48) Further increase of y will result in T, becoming negative, i.e. becoming a compression. If after this bas occurred, F is caused to diminish, ultimately becoming a compression, it is easily seen that T, will always remain negative, as friction and slope @ will now be assisting each other. No matter what compression may correspond to 'T’,, y cannot he caused to diminish. Thus the value of y given in (43) really corresponds to the beginning of permanent set. For values of F beyond that given by (43), complete recovery cannot be afterwards attained. We may therefore call this value of F the elastic limit. It is to be distinguished from the value of F obtained from (9) and (40), viz. he (Ay + Zn») UN, , Xo This latter limit corresponds to the limit of Hooke’s law. For values of F less than this, Hooke’s law will he ac- curately followed, and there will not be any hysteresis effects. If F be increased continually beyond the value given by 2M 2 (44) 532 Mr. 8S. Lees on a Simple Model (43), not only will T, become negative, but dT./da will become negative and numerically increase with @ or y. This may be seen at once physically, or may be verified by the algebra of (41) and (42). Thus a time will in general arrive when dI',/da= —4d,, and when this is the case, dec an ann This would correspond to yield point, as ordinarily under- stood in the testing of materials. With regard to the term (No—y’) which has been fre-_ quently utilized in this section, we can suggest a rough interpretation. It is essentially to be regarded as a force of cohesion. Hence the value y = N,/8 must be regarded as a slip sufficient to break down this cohesion. Thus it may roughly be identified with the slip producing /racture or failure. In this connexion, it should be borne in mind that the model is only a rough representation of what is essentially a statistical problem, involving a multiplicity of surfaces, whose normals and corresponding cohesive forces act in all directions. § 11. Cyclic State for Modified Model. Teliee., UL. We proceed to discuss the nature of the hysteresis loop for the model when in the cyclic condition. The appropriate diagram is shown in fig. 11. The points L and J refer to to Illustrate Elastic Hysteresis. 533 the states (Fj, x), (—F,, —.x,) respectively ; whilst the cor- responding slips are y, and —y, respectively. We shall suppose that y; is less than y/2a, so that the cyclic condition is possible. By (42), we shall have ral: — 2aNo) (Agvi—PNo) _ yy — aut Soper are : 2u8 —— 0 (45) F, is then given by BE, = Aye} + 2ro (vy — 71) = (Ay ue 2p) ty = 2roY1- (46) Now imagine F diminished, corresponding to the path LM. At first, we shall have a Se asdeswele en (20) and this will go on till slip occurs at M. If the value of « at M be ay, the value of T, there will be (Ts)ar= (No— By2”) (4 2ay1) —Ae(4i1—2m), - (47) which must be a compression sufficient to cause slip. Hence also (Ts) = (No— Ay?) (w+ 2a)... (48) From (47) and (48) we get Ao(@:—em) = w(No—By’). - « + (49) For a given a, (45) determines y;, and then (46) gives Fy. From (49) we can find xy, and the appropriate value of EF at M is easily obtained, since the point lL and the slope of LM are both known. The points J and K are got by symmetry from L and M respectively. To get the shape of the curve KL (and therefore by sym- metry JM), let at any point of the curve the slip be y._ Then (42) holds and gives y in terms of 2, say at Also hence on eliminating x between f(x) and I’, we get the required '—w relation for the curve KL. Without going into the algebra, it is quite clear physically that the shape must be something like that shown. The area of the. complete loop is most easily obtained by observing that energy is lost on the whole, only during the 534 Mr. 8. Lees on a Sumple Model process of slip. Along KL the loss by friction is therefore Yi yy 2(" tty = 2)" By —Rady - GL) a aya! Y1 whilst along the path MJ, the loss is Twi Tu 2 Tody =— 24 (N,—By?)(wt+2ay)dy. (52) 4 yn By changing y into (say) —u in (52), we see that (51) and (52) give the same result, as might be expected. Hence the total loss by friction in describing the loop is Yi : a4? Ay? \(u—2ay)dy = Bun (Me— Fue). (63) WN This is the required area of the loop. It clearly vanishes when the slip y; vanishes. When the slip y; is small, a first approximation for yj is from (45), = (Agv,— No) / (Ag —ZaNo). 5 ‘ 3 (54) A nearer approximation is therefore le (Apt —pNo)? J a8 No = 2aNo av HB (Ag == 2aNo)° 7 ; : (55) Hence from (53) and (55), as a, and y, increase, the area of the loop increases at a greater rate than (Asv,—pNo). This result may be compared with the results of $5, and the comments in § 6. §12. Asymmetric Cyclic condition for Modified Model. We can work out the case of a hysteresis loop produced with unsymmetrical stress limits, following the procedure of $7. A typical resulting loop is shown in fig. 12, which shows a case with the mean stress Fy positive. The point L will still lie on the dotted curve NL of fig. 11, whilst the point J will also lie on the dotted curve JP of the same © figure. The new curves KL and MJ will not, however, be symmetrical about the origin O. As FE, is increased, the point J will ultimately coincide with P, and further increase of F will result in J lying on the curve PT, which is the continuation to the right of P, of the curve JP. A limiting case of elastic hysteresis will arise when Fy is increased to such an extent that JK just touches the curve JPT. This to Illustrate Elastic Hysteresis. 535 will, of course, correspond to an amount of slip given by equation (43). Further increase of Fy will clearly involve permanent set. In the above argument, it is supposed that the variations of F) have been made with the range 2| f,| of the alternating stress kept constant. Fig. 12. F The expression for a typical area of loop in the fgeneral case is complicated, and is not given here. For small slips y, it is clear that the results of calculation for our original model can be used. § 13. Mean-stress-strain Loop for Modified Model. Following the argument of §9, we can consider for the modified model the effects of superposing a periodic slow variation of mean stress on a rapidly alternating stress. There are three cases to consider — (a) Starting from the neutral condition, provided | Fo| and | I,| are small enough, the mean-stress-strain curve will be a portion of the straight line PON of fig. 11. The limiting points of this line will be given exactly as in (a) of § 9. : (6) With | F,| less than Fy, but | Fo} + |F,|> Fn, we shall get hysteresis loops of the mean-stress-strain type arising. Referring to fig. 13, a typical loop like J,K,L.M, will consist of lines J.K, and LM, parallel to the straight line PON, and two curves K,N,L, and M,PiJ>. The curve K,N,L, is clearly obtained as the locus of points G@ such that GR drawn upwards and parallel to PON from G to R (a point on the curve NL) is exactly equal to |F,|. The BAG ie): Mr. §. Lees on a Simple Model curve M,P,J », which is symmetrical with K,N,L, about the origin, is similarly obtained from the curve PJ (compare with fig. 7b) *. Fig, 13. (c) With |F,|>Fy, we get hysteresis loops whatever be the value of ||. The mean-stress-strain-curve, 7. e. the locus of the point given by the mean of the extreme stresses and the mean of the extreme strains for an instantaneous loop with the given | F',| corresponding to any values of Fo, is therefere a curve like UOV, of limited length (see fig. 12). Provided permanent set does not take place, the Jength of UOV will be determined by both of the values of Fol and |F';|, and not merely by |Fo|, as in §9 (c). A limitation is set up to the value of |F)|+/F,| by the restriction that no permanent set shall take place. The algebraic details are not given here. § 14. Application of Model to Shear Stresses. The model of §3 and its modification of §10 can also be used to illustrate elastic hysteresis for shear stresses. Referring, e.g. to fig. 3, we have only to imagine the force F now applied to B, instead of A,;, and the equal and opposite force applied to C, instead of As, to get a model showing shearing action. Weare here neglecting the con- sequent tilting of the portions of the model ; this tilting may be avoided by supposing the component parts to be con- strained to move parallel to the forces F by suitable friction- less guides. Whilst it is not intended that any comparison * Notice, however, that our theory would make the dotted line N’L’ of fig. 76 always coincide with a curve of the mean-stress-strain diagram, which apparently is not the case. a5 to Illustrate Klastic Hysteresis. 537 between the cases of direct and shear stresses shall be made numerically, it is quite clear that such a model will lead to exactly the same type of results in the two cases. A com- parison of the preceding theory with the results of Mr. F. H. Rowett’s experiments is satisfactory in many ways. These experiments dealt solely with shear stress hysteresis (see Proc. Roy. Soc. A. vol. Ixxxix. p. 528 et seq.). $15. Time Effects. It is quite clear that the model and its accompanying theory do not explain what may be called teme effects in elastic solids. By this phrase is meant here the changes which take place in a solid with lapse of time, with or without the application of external stress (kept coastant). Some account can be rendered of these effects by assuming that when slip takes place, the frictional force of slipping is not a constant as in the model of § 3, or a function of the slip as in the modification of § 10, but depends also on the velocity of slip, not on the rate of change of the strain. The difficulties arising out of time effects have been avoided in the above discussions as far as possible. They are re- garded as arising out of change of mw with velocity. ‘This “coefficient of friction” will doubtless depend not only on the velocity of rubbing, but also on temperature, which in turn is a function of the number of hysteresis cycles described, ete. In this connexion, it should be observed that the introduction of w into the argument of the paper is essentially an artifice which brings out the striking analogy which seems to hold between the every-day phe- nomena of solid friction, and the more subtle friction obtained in elastic hysteresis experiments. Conclusion. In this paper the author has described.a simple model which would seem in many ways to bring out, at least quali- tatively, many of the general effects observed in connexion with elastic hysteresis. The ultimate standpoint is that such effects are due to something analogous to the ordinary solid friction of every-day life. ‘he model enables some simple deductions to be drawn of a mathematical character, but no attempt is made here to stress these deductions unduly, as the model (involving one dimension of displacement) can only but roughly represent what is essentially a question of statistics, involving slippings in all conceivable directions. XLIX. Atomic Hydrogen and the Balmer Series Spectrum. By R. W. Woon, Professor of Experimental Physics, Johns Hopkins University *. ; Part I. Atomic Hydrogen. ie the present paper an explanation will be given of practically all of the very curious spectroscopic phe- nomena observed with very long vacuum tubes containing hydrogen, which [described in two previous communications (Proc. Roy. Soc. vol. xevil. ; Phil. Mag. vol. xlii. p. 729). The explanation has apparently been tound of the necessity for water vapour or oxygen for the development of the Balmer spectrum—a matter which has always been a mystery. Very remarkable effects have been obtained, which show how important is the réle played by the wall of the tube and the gas film adsorbed on it. It has been found possible to pump practically pure atomic hydrogen gas out of the discharge tube and study its chemical and physical pro- perties. Wires of certain metals and certain oxides, when introduced into the stream of atomic hydrogen, come spontaneously to incandescence and cause the formation of molecular hydrogen. To make the present paper intelligible, it will be necessary to recapitulate very briefly the subjects taken up in the two earlier communications. It was found that if a hydrogen vacuum tube of moderate bore (4 to 6 mm.), and a metre or two in length, was excited by the discharge of a high-potential transformer (or the direct current from a battery: of dynamos), the central portion of the tube showed only the lines of the Balmer series, the secondary spectrum appearing only in the vicinity of the electrode bulbs at the ends of the tube. By using the central portion (suitably bent) ‘ end-on,” the series was photographed to the 20th line—a gain of eight lines over previous laboratory records. The hydrogen was introduced moist from an electrolytic generator through a long fine capillary, and continuously withdrawn from the tube by a mercury pump. With small currents (0°5 to 1:5 amps.) in the primary of the transformer, the luminosity of the tube was low, and the secondary spectrum predominated, the Balmer lines being very weak. As the current was increased, the Balmer lines increased, and the secondary spectrum de- creased in intensity, reaching its minimum value with a * Communicated by the Author. Atomic Hydrogen and the Balmer Series Spectrum. 539 current of 15 or 20 amperes in the transformer. The actual intensity of the secondary spectrum (at its minimum) was about 1/50 of the intensity which it had at the ends of the tube. As we shall see later on, the probable explanation of the peculiarities thus far outlined is as follows :—The secondary spectrum is emitted by the hydrogen molecules ; the Balmer lines by the atoms. With a heavy current in the tube, the dissociation into atoms is nearly complete and permanent, no appreciable recombination occurring during the very brief current- pauses which occur when the transformer potentials are near the zero value, the duration of which can be observed by viewing the discharge in a revolving or ‘ wabbled” mirror. This time is of the order of 1/1500 sec., and, as we shall see presently, about 1/5 of a second is required for the recombination of the atomic hydrogen. At the ends of the tube, molecular hydrogen is con- tinuously supplied by the bulbs, the metallic electrodes acting ascatalyzers causing the instantaneous recombination of the dissociated hydrogen. The probability of the truth of this explanation will appear when we come to the subject of the action of metallic wires on the discharge. With a feeble current, however, atomic hydrogen is not formed rapidly enough to permit of high concentrations, and the secondary spectrum predominates in consequence. _With the heavy current, about 1/50 sec. is required for the dissociation of all of the molecular hydrogen in the central portion of the tube. This we know from the study ne oe duration of what I called the “‘ secondary spectrum ash.” It was found that with the tube operating under such conditions a direct-vision prism showed only the Balmer lines in the central portion, the intervening regions being quite black (the “black stage’’) ; if the current was interrupted for a moment and then turned on again, the secondary spectrum appeared as a brilliant flash, which lasted from one to three or four half-cycles of the current, according to the pressure of the gas in the tube. At high pressures the duration ef the flash was longer, as was to be expected, more time being required for the complete breakdown of the gas into atoms. The further extension of the Balmer spectrum depends upon ascertaining the cause of, and abolishing, the secondary spectrum and the faint continuous background, and giving a sufficiently long exposure. The most promising line of 540 Prof. R. W. Wood on Atomic Hydrogen attack appeared to be a study of the “Infected spots” which I spoke of in the earlier papers. These are portions of the tube in which the discharge appears white or pink, in contrast to the fiery purple exhibited by the remainder of the tube. The spectrum of these spots shows the Balmer series only to the 12th or 14th member, and a fairly strong secondary spectrum. I suspected that they were due to a. contamination of the wall of the tube; and if the cause could be found, it might be possible to improve in some way the condition of the rest of the tube, and so obtain a more complete series of Balmer lines. The clue was obtained, as I have shown in a recent paper in the Proc. Roy. Soc., by the accidental entrance of a speck of sealing-wax into the discharge tube. This was speedily changed to a spot of stannous oxide by the heat of the discharge, and examination of the spot with a lens showed that it was covered with minute globules of metallic tin. A section of tube was now fine-ground on the inside with carberundum, and this ground portion made a part of the long hydrogen tube; the portions to the right and left of the ground portion were purple, and showed the pure Balmer spectrum, while in the ground portion the discharge was white, and showed a very strong secondary spectrum. The ground-glass tube was tried as a result of an experi- ment with a tube of unglazed porcelain, inserted at the middle of the tube, which was then bent at two right-angles, so that the discharge in the porcelain tube could be viewed or photographed “end-on.”’ The porcelain tube gave only secondary spectrum with the first four or five Balmer lines. A tungsten wire was then inserted in the tube at a spot which showed the pure Balmer spectrum. ‘The wire was raised to a white heat by the discharge, though a fine thread of soft glass inserted in the same way was not even softened. In the vicinity of the wire the secondary spectrum came out strong, and further experimenting showed that the secondary spectrum appeared a second or so before the wire became incandescent. Addition of oxygen to the hydrogen suppressed the heating of the wire. As I have shown in the paper previously alluded to, the action of the wall of tie tube at an infected spot, the speck of stannous oxide, the tungsten wire, and the ground-glass surface appears to be a catalytic one, these surfaces causing a recombination of the atomic hydrogen, thus furnishing molecular hydrogen at a rapid rate, the breakdown of which by the current causing the secondary spectrum. and the Balmer Series Spectrum. 541 In all of the recent work, tubes of pyrex glass, carefully cleaned with hot chromic acid, have been used. The tubes usually get into good condition after a few minutes’ operation, and show less luminosity through a green ray-filter (which is opaque to the Balmer lines) than the tubes of soft glass used in the earlier work. One of the most discussed problems in spectroscopy is why the presence of water vapour in the hydrogen enhances the Balmer series and suppresses to a great degree the secondary spectrum. As I showed in the earlier papers, if dry hydrogen is employed in the long tube, fed in at intervals through a palladium tube and pumped out with the tube in operation, the discharge eventually becomes white, and all of the Balmer lines disappear except H., which is so faint that it appears of adull brick-red colour, in contrast with the secondary spectrum. In view of what we now know, it appears as if the glass wall of the tube, when thoroughly freed from adsorbed water vapour or oxygen, acts as a powerful catalyzer of the atomic hydrogen, which never reaches a sufficient concentration to cause the Balmer spectrum to appear. Dr. Irving Langmuir, with whom I discussed these results, made the very valuable suggestion that the glass surface might be “poisoned” by the oxygen. He has found, in the course of an extended study of the production of atomic hydrogen by an incandescent tungsten wire, that the presence of small traces of oxygen prevented the formation of atomic hydrogen by “ poisoning” the cata- lyzing surface of the metal. This being the case, the oxygen must also render the surface of the (comparatively) cold tungsten incapable of bringing about the recombination of the atomic hydrogen, as had been found to be the case. This makes it appear extremely probable that the part played by water vapour in bringing out the Balmer series is merely that of supplying a “ poison” (oxygen) for the catalyzing wall of tube, thus permitting a high concentration of atomic hydrogen in the tube under the action of the heavy discharge. With a feeble discharge, as I showed in the earlier paper, the secondary spectrum predominates and the Balmer lines are weak. This is probably due to the fact that the atomic hydrogen is not formed fast enough to get ahead of the catalyzing power of the tube wall. If dry hydrogen is admitted through palladium, and the tube brought to the white stage by long operation, it is found that if a condenser is put in parallel with the tube, the discharge becomes red and the Balmer lines appear. 542 Prof. R. W. Wood on Atomic Hydrogen The action of a condenser is to pass currents of enormous magnitude but of very brief duration through the tube. With these very heavy currents (hundreds or even thousands of amperes) we have a sufficient momentary concentration of atomic hydrogen to bring out its charac- teristic lines, even with the tube wall thoroughly freed from water vapour—the condition which gives the secondary spectrum only, owing to the powerful catalyzing action of the walls. Merton’s observation that the discharge became white when the hydrogen tube was immersed in liquid air, is at once explained by the greater catalyzing power possessed by the wall at a low temperature. Langmuir found that the atomic hydrogen produced by incandescent tungsten would not pass through a tube cooled by liquid air, though it passed to a considerable distance down a tube at room temperature. This observation I have confirmed in the study of the properties of atomic hydrogen, pumped from a discharge tube operated by a heavy current, small particles of thorium oxide being brought to incandescence at a distance of 20 cm. from the discharge tube. Touching the wall of the tube leading to the pump with a pad of cotton wet with liquid air, at once extinguished the specks of thoria in the tube beyond the cooled spot. A fine tungsten wire, inserted in the tube leading to the pump at a distance of 4 or 5 cm. from the discharge tube, is brought to a red heat by the current of atomic hydrogen pumped from the tube. Clean aluminium foil of the thickness of writing-paper, when introduced into the discharge tube, at first caused the appearance of the secondary spectrum in its vicinity, but after several minutes’ operation the deep purple colour returned and the secondary spectrum disappeared. If the hydrogen current was now shut off and air admitted, and the tube operated with air at about 0°5 mm. for a few minutes, it was found that, on again operating it with hydrogen, the aluminium had regained its catalyzing power, and the white discharge appeared in its vicinity ; in a few minutes this disappeared, however, as before. This makes it seem probable that slightly oxidized aluminium will catalyze the atomic hydrogen, but that the clean metal will not. The mystery of why the long tube gives a pure Balmer spectrum at the centre now appears to be explained. The more or less oxidized aluminium electrodes act as catalyzers supplying molecular hydrogen continuously from the atomic hydrogen formed by the discharge, and the concentration of the atomic gas never reaches a high value at the ends of the and the Balmer Series Spectrum. 543 tube near the electrode bulbs. In one tube of pyrex glass which had been very carefully cleaned with chromic acid, with especially clean bright electrodes, it was found that the white discharge extended to a distance of only 3 or 4 cm. from the bulbs, while in the earlier tubes it often reached to a distance of 20 or 30 cm. In the earlier papers I have drawn attention to the circum- stance that if the hydrogen tube is brought to the white stage, and then highly exhausted, with the current shut off, if a small amount of air or nitrogen is admitted, the discharge is of a most beautiful golden-yellow colour, resembling the discharge in pure helium. Photographs of the spectrum showed that the second positive spectrum (violet and ultra- violet bands) was nearly absent, the yellow colour being due to the first positive spectrum, consisting of red, yellow, and green bands. Applying the catalysis theory to this result, the indications are that the first positive spectrum is due to the nitrogen molecule, the second to the atom. It was found that the yellow discharge was obtained only if a very small amount of air was admitted, doubtless due to the fact that if too much air was introduced there was enough oxygen to poison the walls of thetube. J have not yet been able to get the second positive spectrum free from the first, but no very great amount of work has been done in this direction. Possibly by intro- ducing an excess of oxygen it can be accomplished. It was found that the tungsten wire was not heated to visible luminosity in the discharge in air, while a platinum wire of the same size was raised to a white heat. Platinum therefore seems to be a catalytic agent for atomic nitrogen, while tungsten is inoperative. Ihave not yet tried platinum in pure nitrogen, and it may be that the oxygen plays a part in the surface reaction which heats the platinum. It seems to be now clear why a more complete series of Balmer lines is obtained in the solar corona and probably in nebulee (provided sufficient exposure were given) than in vacuum tubes. The luminous gases are in these cases not in proximity to catalyzing surfaces, and consequently atomic hydrogen of 100 per cent. concentration can exist. Of course the possibility of a different type of excitation still remains. To further extend the series in the laboratory, it will probably be necessary to devise a method of more com- pletely poisoning the walls of the tube, or abolish the wall entirely, as can be done perhaps by means of a very powerful discharge of the ring type excited by high-frequency in- ductive effects, in an electrodeless tube. D944 | Prof. R. W. Wood on Atomic Hydrogen Pyrex glass has been found better than soft glass, but quartz appears to be no better than pyrex. i The fact that we have only atomic hydrogen in the central part of a long spectrum tube, even during the brief moments when no current is passing (between the half-cycles), makes it appear probable that the refractive index of the gas in the atomic condition can be determined by introducing the tube into one path of an interferometer, and illuminating the instrument with flashes of hght, during the moments when the current is not passing, by means of a disk perforated with two slots rotated by a synchronous motor. This experiment will be tried in the autumn. Part il. The Balmer Series. In continuing the work on the Balmer spectrum of hydrogen, tubes of pyrex glass have been used exclusively. © The aluminium-foil electrodes were attached to tungsten wires, which fuse easily into pyrex. As there is apt to bea slow capillary leak along the tungsten wire, a drop of sealing- wax was always melted around the wire on the outside of the bulb. These tubes, if made of carefully cleaned glass, will come into the ‘‘ black stage’ (showing the Balmer lines on a black background when viewed through a direct-vision prism) after ten or fifteen minutes’ operation. The lines of the Balmer series were photographed in the 2nd- and 3rd-order spectrum of a very perfect 7-inch plane grating (temperature controlled to 0° 1 by a thermostat) with an objective of 20-foot focus, H,, Hs, and H, all showing as clearly separated doublets. The series was recorded with this apparatus as far down as the 18th line with an exposure of only 12 hours. Only a few minutes were required for the plates showing H,—H,. Ourtis, using a concave grating of rather short focus, with an exposure of 5 hours obtained only the first six lines of the series. When we consider that the sixth line has an intensity about 4000 times as great as that of the 18th line, the enormous intensity and efficiency of the long end-on tube is at once apparent. The tube was excited by a large 6000- volt transformer. | Photographs were also made in the 5th-order spectrum of a plane grating, in combination with a collimator and Cooke portrait lens of 1-metre focus. A screen of glass coloured by nickel oxide was used to cut out the overlapping green of the 3rd order and violet of 5th order, which covered the region of the last lines of the Balmer spectrum. This glass is opaque to all visible light except the extreme red, and is <4 and the Balmer Series Spectrum. 549 highly transparent to the ultra-violet region in the vicinity of the end of the Balmer series. The grating was selected by illuminating it with the light froma quartz-mercury arc, mounted in a closed box provided with a window of dense nickel-oxide glass. This arrange- ment gives a powerful beam of radiation of wave-length 3660, which was rendered convergent by a lens and reflected from the grating to a screen of barium platinocyanide. In this way a grating was found which was enormously bright in the 5th-order spectrum for the region of the end of the Balmer series (W\=3676). With this grating and the new tubes of pyrex glass, the 20th line was photographed with certainty, and probably the | 22nd line. The 21st is so nearly in coincidence with a strong line of the secondary spectrum that it cannot be identified under present conditions. H, and Hg were not photographed, as we have Curtis’s values, which are of the same order of accuracy as those about to be given. The plates were measured and the wave-lengths computed by Mr. Arthur E. Ruark, one of my students, to whom I am indebted for a large amount of very faithful and accurate work. The reference iron-lines were the tertiary standards determined by St. John and Babcock ™*, lines showing no pole effect being used in most cases. Reduction to vacuum was done by the table of Meggers and Foote fT. Difference between these values ai hier Ni tvels and these of Curtis. eee *6562°793 0 eee *4861°326 0 : 4340-465 4341-681 —0:001 a ee 4101-731 4102-884 aU Vi Sab get ce [oe 3970078 =: 3971-192 SeOOR (Ske see an, Cees 3889-064 3890°161 TSANG A he eect ies osu. 3835°397 3336°481 “ea O10) 2 eee 3797-910 3798°984 + 010 - pose note 3770°634 3771°701 22-00 | 1 37507152 751-214 + 902 oe 3734371 3735°429 ‘000 | es: 3721-948 3723:008 + :007 From Le Spigenenee 3711-980 3713-032 a OUR | eet luted Were). 3703°861 3704-911 re CLG Gun ai. Le eS... 3697°159 3698-197 + -005 | Ve ett 1G pies 3691°553 3692-600 + 004 | heen 3686°833 3687°878 2008 rt Partita 3682825 3683-869 + 015 | 19 Foes 3679°372 + O15 | eee 3676'378 “01s | * Values given by Curtis. * Astrophys. J. vol. liii. (1921), or Mt. Wilson Contrib. No. 202. + Bur. of Standards Sci. Papers, No. 327 (1918). Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2N 546 Mr. D. Coster on the Spectra of X-rays Ov one of the plates, taken with an exposure of only ) minutes, H, appeared beautifully separated, the distance between the components being fully three times the width of either component. This makes it appear probable that we can resolve also the 4th and possibly the 5th line, by operating the tube with a current of comparatively small intensity. | Two series of measurements, each consisting of fourteen settings on the components of Hy, gave the following values for the doublet separation : TSGHSORTCS 5 ey cee ey ahead "0568 Pinduseries: | fri (76 Wane Toittee 0601 Mioain jae aateeee 0584 J.A., which is in exact agreement with the value obtained by, Gehreke and Lau and is ‘004 I.A. less than the value given by McLennan and Lowe. The actual wave-lengths of the components of the doublet H, are 4340°494 and 4340°435, the value given in the table being, of course, the centre of the doublet. I have been aided in this investigation by a grant from the Rumford Fund of the American Academy. Baltimore, U.S.A., June Ist, 1922. L. On the Spectra of X-rays and the Theory of Atomic Structure. By D. Costsr*. Part LV.f New Measurements in the L-series of the Elements La (57)—Lu (71). iL N Part II. of this paper I gave the results of my measurements of the elements Rb—Ba, in this part I will deal with those of the rare-earth metals. As is known, the latter elements are very similar in chemical properties, * * Communicated by Prof. Sir E. Rutherford, F.R.S. + See Phil. Mag. xlii. p. 1070 (1922), Parts I., IL., and TIT, aoa and the Theory of Atomic Structure. 547 and it is rather difficult to get them in a very pure state. Some preliminary investigations with rather impure speci- mens of these elements showed clearly that from these no certain information could be obtained about the weaker lines, since the great number of ‘“ foreign ” lines appearing on the plates make the interpretation of the photographs nearly impossible. The present investigation has been rendered possible by the kindness of Mr. Auer von Welsbach, who offered to this laboratory a beautiful collection of very pure salts of these elements. Ou the photographs taken with these salts usually no lines were observed belonging to any other rare-earth metal. As the presence of about 0°1 per cent. of another element would be sufficient to give the stronger lines, we may conclude that the salts used were extremely pure. In the ene ease, however, of Yb (70) several lines were observed belonging to Lu (71). §2. The apparatus was the same as that used for the former work. In this region of wave-lengths the spectro- graph need not be exhausted. The slit of the X-ray tube was covered with an aluminium sheet of 104. After being glowed in a Bunsen flame the salts (sulphates) were pressed ona roughened copper or silver plate, which was soldered on the anticathode. Imperial Hclipse plates were used, which appeared to be much more adapted to X-ray work than the technical X-ray plates previously used in this laboratory. These plates are extremely sensitive, especially. for wave-lengths of more than 3A.U., and even for wave- lengths of 1-2 A.U. they give betier images than the technical X-ray plates in half the time of exposure. For the rather small glancing angles in question (10-20 degrees) the apparatus seemed in the beginning to be very inconstant, giving sometimes fairly good and sometimes very bad plates. This phenomenon appeared to be connected with the reflexion on different parts of the crystal; very good plates were only obtained if the radiation was reflected by the middle part of the crystal. As it seems to be impos- sible that this phenomenon should depend on the geometrical conditions of the apparatus we must assume that the reflect- ing power of the crystal was greatest at the middle part. Once the cause of this phenomenon was known, it was only a matter of time to obtain rather good plates for all the elements, as by a change of the position of the focus spot on 2N 2 hy 548 Mr. D. Coster on the Spectra of X-rays the anticathode the radiation can be directed on any desired part of the crystal. This change of the position of the anticathode spot was easily obtained by a change of the rela- tive position of the anticathode and the hot-wire cathode. The hot wire must be renewed about every 8 hours, and frequently after this operation it was necessary to readjust for the most favourable conditions of reflexion. The tension on the tube was about 25 k.v., the current was not more than 15 m.a., the time of exposure varying from 10 to 45 minutes. Calcite was used as analysing crystal. Some trouble was caused by the copper K lines and the tungsten L iines, which cannot be avoided when working with a tube of brass and a tungsten hot wire. In a few cases these foreign lines made it impossible to measure some weak lines belonging to the element under inves- tigation. § 3. As regards the accuracy of the measurements I may refer to Part II. $4 of this paper. In general, the lines for the elements La—Lu are much sharper than those for the elements with lower atomic number. On the other hand, most of the stronger lines are accompanied by fairly intense satellites which often lie very close to them, thus diminishing the accuracy of the measurements. Usually the errors in the wave-lengths of the satellites themselves are larger than those in the wave-lengths of the diagram-lines, as the satel- lites are more diffuse. . With the exception of the lines 6, and yg, which are dealt with in Part IV. § 10, all the diagram-lines (see diagram IV. Part V. §2) are given in Tables XIII., XIV., and XV. The lines appearing with two decimal places have ‘been taken from Hjalmar’s precision measurements*, the other lines having been determined in the present investigation relatively to these lines or to the copper, tungsten, or zinc lines, which also appeared on the plates. For the elements Hu, Gd, and Tb the line 7 could not be separated from a, or #, which are much sharper and more intense. * Zeitschr. f. Physik, iii. p. 262 (1920) and vii. p. 841 (1921). O49 Structure. ? 4 ic and the Theory of Atom L-1PIL 0-681T G-ELGT L-6161 FILET 6-6¢F1 8-181 6-€091 8-0PLT 6-G181 G-C68T L-8L61 G-LLUT 8-616T 8-TTéT 6-198 6-SLFL €-89F1 6-Gé¢T L-L8ST L-T99T ¢-G6LT 6:698T 6-096T 9-9606 fh G-E8IT 9-S66T P-8IE1 L- LOST 6-0GF1 8-ELF1 0-TE¢T 6-€641 6-S¢9] P-L6LT 0-GLZ81 6-61 9-TF0G zd €.0881 8-591 | ¢-z9et | &PIPI L.69FI 9.9281 29-881 FOOT 60-8LI 88-8181 18-961 88-FF02 06-LE1Z _—_—_—_— th 9961 0-S061 SOFT 6971 GEIST G-FLGT 9-LE9T GOLT L-GL41 §-1&61 T-9106 9-GOTS 8-0066 ch G:LOSl | &PIFl S-GIFl | L-GOFL 9-011 | 9-89¢T L-89S1 | §-819T R-6191T | L-LL9T 06491 | G-LELT 6-1PL1T | 1-S0ST GS808T | G-OL8T T-8181 | G-GP6I F-180G | §-660G 8-F11Z | 6-S816 T-F0GZ | 6-916GG 0:8666 | 6-ELE6 “| ag 68681 v-6PF1 6-L¢¢T 0-9T91 L-LLOT ¢-GPLI 6-O18T L-88T 0-846T GGGIG b:G1GG 64086 6-G0PG tg L-0GF 1 G-GLFI PP-E8ST GS-F91 8¢-90L1 89-BLL1 OF-GPSI 18-9161 1G-£661 16-29% 06-8922 00-TS8% O8-ESFZ 9) G-LEFL G-88Fl F-96971 §-99T L-OTLT F-18LT €-6F81 1-661 F-9661 G:G9IG 1-0966 GFPEG 8-EPFG rg) *("WO 17_OT) “Q'X Ul Syycuey-oar AA “TITX @Tav T¢-CT9T 6L-L99T OF-0841 86-0F8T 09-F06L 6F- TL6L 66-1F0G 66-9116 10-G61Z T§-C9&6 OL-LEFS 00-9996 89-6996 96-9691 6-8L91 OF-16LT 90-6931 P9-ST6L 1€-G861 69-6206 SE-LGIG 89-406 €9-GLE6 89-LOFG IT-¢9¢¢ 66-8996 l 8-1§8T 0 068T 1-G10@ 1-6806 0-F916 0-6666 T-LO€G $0686 0-LLFG €-0L96 L-8LLé 0008 OTE "OK OL | 1 89 “OH 19 aL o9 “PD FO) nT 69 | US GQ | “PN 09 | Id 69 “*aQ 8g “eT La —— ) Mr. D. 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Coster on the Spectra of X-rays § 4. The classification of the lines proposed in this paper is strongly supported by Tables XVI. and XVII. Table XVI. gives the relativity L-doublets. From this table we TABLE XVI. Relativity L-doublets. | | ge! By-%, | Ys-Be %1-B2 | Yo-Pis | Lp- Ly gules ne | 29°51 | 30:01 | 80:19 | 29-86 ie yes 58 Ca lee | 14 ME 82:85 9825" | 82:20 | So AoNmeaD /59 Pr.......... 85°53 | 35:02 | 35:10 | 3479 | 34:80 | 35:24 60 Nd......... 8177. | «87°86 |) ST 76 |. B71 1 Br6T | aemee 62'Sim 4. | 48°70 | 48°95 | 44:17 | 48°64 | 43-62 ~ 68 Blanes: e 47-17 | 47-41 | 4688 | 47-10 ie 64 Gd.....2. ne 50°64 | 51-11 | 50-46 | 50:83 x Goby. ana Ke 54°36 | 5448 | 5419 | 54:32 a 66 Dy......... 5852 | 5827 | 5827 | 57-43 rf G7 AKon gil 62.48 | 62:43 | 61:74 | 61:58 | 62-08 68 Eis: 67:09 | 66:80 | 66:88 | 65°68 F 70. Nii. 7658 | 76:09 | 7636 | 75:05 oe Tilman ee | 8156 | 81:12 | 8088 | 80-22 7OVRaya ee. | 93°33 | 99°68 | 91:99 | 91-64 TAN ee | 98°76 | 98:54 | 99:00 | 98-05 TaBLE XVII. Differences in A R: OS [Sr Oy Were eR Bila eet 1054 | 0-192 | 0779 | 0:504 FASS OB ck soca 1:048 | +192 "782 500 5OuPr so 4 1:059 | 192 ‘786 -496 60Nd eon) 1:060 | +193 ‘795 495 62iSmmes 1076 | -194 ‘801 49) 63Eu..... ... As 195 ‘790 ‘487 64 Gd... 193 ‘807 -488 Gorin - 195 ‘806 ‘481 66 Dy......... 1:095 | +198 806 ‘487 67 Ho. 1099 | -198 819 -488 68rd ee 1104 | :196 “802 494 70 Yb. 1:108 | +198 822 | -498 FAT oe 1118 | 202 ‘816 499 (alata 1119 | +204 -848 “494. GAN ta 1:136 | -202 "829 483 BBeR iy. Kan 1-200 | 224 910 ‘495 may conclude that the frequency difference y,—, is essen- tially smaller than the other doublet differences * Compare Part II, § 5 of this paper. Physik, vi. p. 185 (1921), table 3. * As the See also D. Coster, Zevtschr. jf: ~ and the Theory of Atomic Structure. D998 lines y; and 8, were usually measured only by reference to other lines this table is not very well adapted for an accurate calculation of the energy difference between the N, and N, level (see diagram IV.). The lines y) and @,, are dealt with in §11 of Part IV. The last column gives the frequency difference between the absorption discontinuities L, and L, as measured by the author. Table XVII. gives those differences of the square roots of the frequencies, which according to Part II. $5 are nearly constant. As the absolute errors in the frequencies are about inversely proportional to the second power of the corresponding wave-length, it is scarcely surprising that there are more irregularities in this table than in the corre- sponding Table V. §5. In Table XVIII. the L absorption discontinuities of some of the elements are given. They were found on the same plates where the emission-lines were photographed TABLE XVIII. Absorption discontinuities. | Wave-lengths. Frequencies. | | | Mg. Vy ils Baan) 2 due 2159°7 | 2005 || 421-94 | 454-44 HOWe Ae oe: | (2158) | (2007) || (4223) | (454-0) 20727 1919-7 || 439-44 | 474-68 59 Pr Spi (2071) | (1922) || (440-0) | (4741) 1990°3 | 1887°6 | 457°86 495-90 GOING 65.3. | (1992) | (1842) | (457°5) | (494°7) Sn eee 18409 | ... || 495-02 | Ge Bun i 1773 Pies BLOG I} | (compare Part II. § 9), and were measured relatively to 8; or y; of the same element. MHertz’s values are added in parentheses ; they agree better with ours for the elements Ce, Pr, and Nd than for Ba and Cs* (see Tables XI. and XII. Part II. § 9). Absorption-discontinuities may be measured very easily in this way, but the method is only applicable if * It may be stated that my values for the latter elements agree very well with those of Mr. Lindsay (compare Part III. § 8). D04 Mr. D. Coster on the Spectra of X-rays no emission-lines are found in the same place, and, again, it seems to be only suitable for a definite range cf wave- lengths. § 6. The faint lines lying on the short wave-length side of B2 gave a great deal of trouble. The photographs of the elements W, Ta, and Lu, as well as those of the elements Sb-Ag, showed that we have to expect 4 or 5 lines in this region. For the elements Er—Sb, however, these lines lie — very near to one another (they all fall in a length of less TaBLeE XIX. Wave-lengths. Bx By | Pro Bir By» AN ee ee I’ (Bs) b 8620) 3680 "|" 366331) Sena PAS OES siecle eau ge ve 3477°5 | 8468-4 49 Mra see, $817 «| 3259°8 | 3265°8 | 8304-0 | 3295-9 5O'Sm one. 3149 | 31081 3114-4) 31426 | 3134-7 GU SIor gy an- | (B,,)° | 2965°8 | 29725 | 2993-4 | 2985-8 SO ees. 2480 | 2478 By | 2483 S6aB ane rae 23756 | 2371-2 a 2381-7 | Bi Way ees 2270) 2297 |. )2285 (Bo. Bao) 58 Cente. 2176°3 | 2184-0 2191-6 | “4 HOUR st ale 2087:-+'| 2095°8 21025 z 60'Nd......... 2004;3))| 2010-7: | 2019:3)| , 162 Simin 1852°3 | 18581. 1865-7 (840) OS Mu eee MSS) Y M6LGid ee 71976: 1 By AI “ WHS D year at a — 1655°8 cy Gan | 4 166,Dy...... 51) 15957 es oe uf | (68 Bie. | 14892 | 14823" (8, | 1501-4 | PAU sess 1345°9 | 18380 , 1339°8 | 1359 Go any poe | 12600 | 12429 | 1250°6 | 1273'8 PAWN elo | 12208 | 12021 | 1209-4 | 1235°4 | | | | than 1 mm. of the plate), so that we may assume that they must coincide or cross in several cases. This made the classification of the lines a very complicated puzzle. Several solutions of it have been tried, that which seems to be the best is given in Tables XIX. and XX. The following facts support this classification. It has been assumed that the line 8; does not occur in this region under the conditions of the experiments, for the following reasons. For the elements in the neighbourhood of W it was shown * that 8; and the L, discontinuity nearly coincide. For the elements from Eu (63) to Cs (55) the L, absorption discontinuity was obtained on the plate as a white line on a * D. Coster, Phys. Rev. xix. p. 20 (1922). --— = and the Theory of Atomic Structure. ddd dark background. No indication of an emission-line which could be 8; was found for these elements. Interpolating between the values of 8; for W and Ta and the L, discon- tinuities of Eu—Ba we get values which do not coincide with a measured line for any of the elements Lu-Gd. Only in the case of Tb (65) a very faint discontinuity in the blacken- ing of the plate was observed at about 1647 X.U., a value which agrees very well with the interpolated value ; but this discontinuity in the blackening looked more like an TABLE XX. Frequencies. | | | B, | By Pro By By» Bo-Byo | %- %. | 47 Ag... (8,) | 251-67 | 25105 | 248-76 | 249-40 | 0:62 | 0-43 AROdeat.. ., | H re 262:05 | 26273 |... 0:50 49In ... 274-73 | 279:55 | 279:03 | 275°81 | 276-48 | 0:52 | 0:57 50 Sn... 289°37 293:19 | 292-60 | 289-97 | 29069 | 0°59 | 0-62 51 8b... (B,,) | 307-26 | 30657 | 304-43 | 305°20 | 0°69 | 0-69 55 Os...| 367°38 | 36858 | °B, | 367-00 1:20 | 1:04 56 Ba... 383°60 | 384:31 en 382-62 i 111 57 La... 401:41 | 400-23 | 398:87 (B5, Bro) 1:36 | 1:18 58 Ce... 418-73 | 417°25 | 415-80 ty 1:45 | 1:26 59Pr... 48656 | 43480 | 433-42 a 1:38 | 1:49 60Nd... 454°65 | 45299 | 451-29 ” t- 70s cer 62Sm... 491-95 | 490°44 | 488°42 | (Bx) 202 | 2:00 63 Eu... 510°76 | 509°78 | 507-31 ob Ap 208 64Gd... 53022} (B,) | 527°32 . Bel 29:38 a 550°34 * 547-63 | : Sie 252 66 Dy... 571-09 i MM re ns 2°75 68 Er... 61191 | 61469 | (G,) 606-96 278 | 3:14 71Iu... 677-07 | 68363 | 680715 670°61 348 | 3-76 73 Ta... 72330 | 733:22 | 72865 715°60 4°57 | 4:37 74W... 74645 | 758-06 | 753-50 | 737-60 4:56 | 4:60 | | | absorption edge than a line, whereas the line 8;, though very faint, for W and Ta is still a very sharp line. § 7. The line 8; follows Moseley’s law closely. Further- more, the behaviour of this line is what we should expect from theoretical considerations (compare Part V. § 2). For Cd and Ag it could not be separated from the broadening on the short wave-length side of 8, but, since for Ag and espe- cially for Cd this broadening extends to a larger region and is more sharply limited than for the other elements in the neighbourhood, we may assume that 8; exists also for these elements. § 8. The lines 8, and B,) were studied by the author in a former paper *. They are both very faint lines, Aj is still * Zeitschr. f. Physik, vi. p. 185 (1921). D956 Mr. D. Coster on the Spectra of X-rays weaker than @,. They seem to correspond to transitions which are inconsistent with the rules of selection given in Part I. § 3. , should correspond to the transition M,—Ls, By) to the transition M,-L;. From this we must expect that the frequency difference of these lines is equal to the fre- quency difference La,—La,. Both differences are given in the last columns of Table XX. They agree within the limits of experimental error. Further, we must expect the follow- ing relation to hold for the frequencies : LB, = L2,+ L;—L,, where L, and L; are the first and third absorption discon- tinuities in the L-series. In Table XXI. the frequencies TaBLE X XI. Frequency of Bo. Calculated. Measured. | AT Ne MER Ss 25195 | 26167 | AS iian nee PPB || Ban HES) + BOSH posse 99312 | 293-19 BSD vate: 30739 | 307-26 OS ane acs 36836 | 36858 56. Ba ye. 38482 | 384-31 Bele: ie: 417-4 Pel HO Pr ie. 435°3 We ASA: S 0m ae 60 Nd......... 453 3 hi 42-995 ip G2 Sma ee 4905 | 49044 | TA We, 75752 | 758-06 | calculated in this way are compared with the experimental values. The absorption edges of Ag have been taken from Part I. $9, Table XII., the edges for Sb, Cs, and Ba have been taken from measurements by Mr. Lindsay, those for W from Duane and Patterson. For In and Sn the frequency difference y,—8; has been taken, which for these elements must be nearly the same'as L;-L, ; for Cd, Pr, Nd, and Sm, L, has been taken from the author’s measurements (see Table XVIII.) ; whereas the value for L; for these elements has been calculated from y, (it has been assumed that for these elements the frequency difference L3;-y, is 2 units). As is seen from the table the agreement between the measured and calculated values is very good. § 9. The lines 8;; and Bi, have recently been measured by the author for the elements Nb-Sb (see Part II. Table X.). and the Theory of Atomic Structure. DDT For the elements Mo-In they are fairly intense lines, but they are not very sharp; their intensity seems to decrease rather rapidly for elements with higher atomic number than In. As has been pointed out in Part IT. § 8, the frequency differences 8,,—83; and {.-8, vary linearly with the atomic number. If we assume, as has been done by Dauvillier * and Wentzel f, that the line 3, found for a great number of the elements Ta—U f is a double line, it is very probable that this line is the same as the lines formerly denoted as By, and fj, for the elements with lower atomic number. The variation in intensity of these lines, however, is very remarkable. As has been stated above, their intensity rela- tive to that of the other lines is decreasing for the elements with higher atomic number than In ; for the elements in the neighbourhood of W these lines are extremely faint, whereas their intensity seems to increase again for the elements in the neighbourhood of Au. Besides, their frequency difference with the lines 8, seems to vary for the heaviest elements -more rapidly than with the first power of the atomic number. This made it somewhat difficult at first to recognize the line 8, of the heaviest elements in the lines 8, and fj, for the elements Nb-Sb. Where two lines were expected nearly to coincide, in most cases a broadening or darkening of the measured line was actually observed. In this case the wave-length given is only a mean value. In Tables XIX. and XX. this value stands in the column of the line which was considered to be the most intense of the two coinciding lines. For Ho (67) and Yb (70) the lines in question could not be measured, as in this region copper K lines appear. It may be pointed out that in general for the elements of Tables XIX.and XX., with the exception of some copper and tungsten lines, no other lines were found on the short wave-length side of f, other than the lines appearing in the tables. | § 10. Table XXII. gives two very weak lines, which were only found for some elements where very good plates were obtained. y; is a non-diagram line, formerly given for the elements Mo—Ba (Table X., Part II.) ; yg is a diagram line, representing the transition O;-L,. As is seen from the table the frequency difference Ly,-L8, is equal to L8,-Le, within the limits of experimental error. The line yg was * Dauvillier, Comptes Rendus, clxxiii. p. 647 (1921). + Wentzel, Ann. d. Physik, \xvi. p. 487 (1921). t D. Coster, Zertschr. f. Physik. 1. c.; Dauvillier, 1. c, (Dauvillier calls this line f,"'). 598 Mr. D. Coster on the Spectra of X-rays Taste XXII. Wave-lengths. Frequencies. Vas Ys Ue Y8 Ya—By B,-2, DO BAG ances: 2218 2218 410°82 | 410°82 | 27-22 | 27-74 Dic) UCIe Aa eae 2029 2019 449°22 | 451-24 | 32°51 32°35 Owe Tee tt. 1942°2 | 1932-2 || 469-20 | 471°62 | 35:06 | 35°02 OO INidiseeie 6: 1859 ies 490-20 ee ok Le Coe De ae 1644 1629 564:18 | 559°23 | 48:47 | 47-17 Gey ee. 1211 1202 10232 | "70802 | 80°96 Sera Hie A aie WAL ER a 1117 ee 815:83 | 92°53 | 92°68 WA Wir a cape ee TOPO aul) Pte: 84456 | 98-11 98°54 never observed by the author for elements with higher atomic number than 74. yy; is not only an extremely weak, but also a rather diffuse line, which for the heavier elements lies rather close to the very intense line y;. ‘This makes it nearly impossible to be sure of the existence or non-existence of this line for the heaviest elements. $11. Tables XXIII. and XXIV. give some lines denoted as B14, Yo, and yo, which were only observed in the region of the rare earth metals. Their appearance seems to be connected with the change in the N-shell of the atom, which we must expect in this region according to Bohr’s theory (compare Part III. $8). For Ba the lines By, and yo do not exist, neither were they observed for La, but here the plates TABLE XXIII. Wave-lengths. Bua Y9 Yio By3 (?) 50 (Cattanine | 2236°9 06 Ba ...,02..2 2140-2 | Bf ua eae Ne us 2048°1 58 Cel. Ste 2212°1 | 2051 1962°3 | SO Pisa ; 21220) 1962°2 ; 1881°1 BOON dd icinacase 20388 | 18804 | 18022... Oe SiMe. 1885°1 | 17285 | 1659°3 | 1987°1 02), eee 1781-4 | 1659°3 oe 1909°2 G4, Gall,....%... 17481 | 1593°6 | | 1835°5 69 Tb 6.0.50; 1685°1 | 1531*4 | 1765°5 GO Wy oo... ad. 162571 ae 1699-2 67 Ho 1567 1416 1635°5 06 Hitec. A 1512 a ama || | 1575°6 | | and the Theory of Atomic Structure. 559 TABLE XXIV. Frequencies. Bis Ys Yio Bo-Bis | Y17Yo Y27Yio Bete * Sa areoe) Uiten eht a O84 56 Ba......... trys (hh, SBBRBHb ® cea, ba: 1-25 Be Ledisi... ats ea MAAAGS D8! ou Tel coc: 1-44 58 Ce ........., 411:95 44437 , 46439 | 1:50 | 1:37 1:44 PEP 0. 42943 46442 48444) 145 | 1:26 15 GOWri...3..... 44696 48463 50564] 1°63 167 1:58 tS 483-40 527-20 54920] 1:82 1:66 1-12 SE | 502708 549718 its 1°89 1:67 JS ae | 521:25 . 572-08 1°89 1°52 BOR ic... | 540°77 | 595-09 1:97 1:84 oo en 560°76 AG 1:83 “i tO... <5. ...| 581-48 | 643°56 1:50 OeOr, 6S Mies... 60270. 0:54. or | | were not especially good. For Ce they are very weak, lying on the long wave-length side of 8B, and y,. From Ce down to Tb they get more intense (for Tb they are of about the same intensity as §,) at the same time as their frequency difference with the lines 8, or y, slowly increases. For the elements with higher atomic number than Tb, these differ- ences rapidly decrease, whereas their intensity relatively to the other lines remains nearly the same. Tor the elements Ho and Er they could not wholly be separated from @, and y, ; for these elements the values of their wave-lengths were roughly calculated from the broadening of @, and y,. For Yb (70) and Lu (71)-8y, and yy had wholly disappeared. For Dy 8:4 could not be observed, as for this element the tungsten La, line was appearing on the long wave-length side of B,. As is seen from Table XXIV., the frequency difference y;—y9 seems to be somewhat smaller than the difference 8,-9 44. A very remarkable line is yo, which appeared for the first time for Cs at the long wave-length side of y.. Itisa rather intense line, being more sharply limited on its short wave- length side than on the other side. As is shown in Table XXIV. its frequency difference from vy. increases first with increasing atomic number, but for elements with higher atomic number than Nd the same difference rapidly decreases, so that this line could not be separated from y, for elements with higher atomic number than Sm. But up to Er (68) Y2 was somewhat broadened and diffuse on the long wave- length side, indicating that also for these elements 79 is still present. For the elements Nd-Cs, where yj, was wholly separated from yp, the latter line appeared to be a faint, but 560 Mr. Ty Coster on the Spectra of X-rays fairly sharp line. Since in the elements Ta—Er, for which Yio 18 not present, a change in the relative intensity of y2 was already observed, it appears that this change in intensity cannot be interpreted as being connected with the presence of 49. For Te and the elements with lower atomic number Yy2 can no longer be separated from v3. It is difficult to say whether 49 still exists for these elements or not, but it seems to be hardly probable. | For the elements Sm—Eu a faint line was observed on the short wave-length side of 6, The presence of @, in this place made it impossible to study this line for the elements with lower atomic number than Sm, and in the same way the presence of $8, made measurements impossible for elements with higher atomic number than 68. It may be that this line is the same as 8,3 previously measured for the elements Rb-Sb—its wave-lengths are given in Table XXIII. Part V. Comparison of the new Haperimental Results with Bohr’s Theory. § 1. The L-series of most of the elements from Rb (87) to U (92) has now been thoroughly investigated by the author™, working in the laboratory ot Prof. Siegbahn ; and from this work it appears that the same simple laws which have been dealt with in Part I. govern the emission of the X-ray spectra of the elements from U down to Rb. There are still some lacune in my tables, but these have no particular meaning, as it should only be a matter of patience and time to fill them. Some rather faint lines were overlapped by sharper and more intense lines of the same or another element, making measurement very difficult. Of some ele- ments no suitable specimens were available, some elements gave difficulties due to their high vapour pressure. In this counexion it may be pointed out that the L-spectrum of mercury has been measured in a very ingenious way by Mr. Miller}; most of his results agree very well with ours. The L-spectrum of tungsten has been measured by several authors with essentially the same results. In addition, Dauvillier | has measured the L-spectrum of U, Au, Pt, Ir, * As has been stated above, for the measurements of the elements Rb-Ta, use was made of the precision measurements of the strongest lines of these elements done by Hjalmar, and for W Siegbahn’s precision measurements were used. For the wave-length tables of Ta—U, see D. Coster, Zectschr. f. Phys. vi, p. 185 (1921). + Miller, Phil. Mag. xli. p. 419 (1921). {t Dauvillier, Comptes Rendus, clxxiii. pp. 647, 1458 (1921), and the Theory of Atomie Structure, 561 and Os, and in the lighter elements Sb and part of the Ce-spectrum. In many respects his results agree very well with ours ; there are, however, also some important diver- gences *. Tt seoms to me that, in general, Mr. Dauvillier attaches too much importance to some lines, which he could only establish for one or for very few elements. Diagram [V.—Niton, XENON, and Krypton, %, j ON ean c ot ee ee ee 8 hy a Semmes Doom ae Ny. eke a a te ne Soe oF TUL 2 a aoa Spee ele eS ae a ee ba a es pr so a . AE es ee i 0 See SP one a a M, iMIAIAIAIBAAMINIAIAIAT ee ea L BAK KH 172,44 Bi neg lsh Pr — oat ST Rimini awe ae Le : a ky poeta) ee J / A a2, 5h, ky A b4 § 2. In Parts I. and III. the level diagrams for niton, xenon, and krypton have been given. These constitute a summary of the measurements, giving at the same time an idea of the successive development of the shells of electrons according to Bohr’s theory. Diagram IV. represents a combination of these former diagrams. The line yg, which now has been established for * Compare D. Coster, Comptes Rendus, clxxiv. p. 378 (1922). Phil. Mag. 8. 6. Vol. 44. No. 261. are 1922. 20 562 Mr. D, Coster on the Spectra of X-rays 7 elements (see Table XXII.) has been added, besides some of the lines belonging to the N-series which has recently been detected by Mr. Dolejsek * in this laboratory have been added. The leyels falling out between niton and xenen are denoted in the figure with one dash, those which fall out between xenon and krypton with two dashes. The diagrams I, II., and III. have been thoroughly discussed in Part III., where a comparison of the experimental results and the theory has been given. Here will only be added something in connexion with the line L8;, which has now been measured for a great number of elements. This line is for all elements rather faint, and becomes still fainter for the elements with lower atomic number than Ba (56). As has been stated in Part IV. § 7, for the elements with lower atomic number than Ag, it could not be separated from §>, which for these elements has nearly the same wave-length as the L,-absorp- tion edge. TABLE XXV, Frequencies. Lp, |La,+Ma,| 1, | i Xs | C6aiyee 2 | 571-09 | 574-27 (74:56) 3:18 | (0-29) Esai 611-91 | 615-73 | (617-03)| 3:81 | (1-30) rag ete 677-07 | 68062 (68330)| 355 | (2-68) rai ee | 72330 | 72612 | 72865) 282 | 258 ayy are | 746-45 | 74914 | 75085! 269 | 1-71 | he Bb 844-93 | 846-76 | 851-281 183 | 4:52 | | are Made od ae 870°77 | 872513 | 877-64] 136 | 551 | CH! eee es 922-64 | 92363 | 93216! 099 | 853 | 59 Ph cf. us 95022 | 95026 | 95954 | O04 | 998 | 9a) 123814 | 123680 1263-20 | —1:34 | 26-40 | Table XXYV. gives the frequencies of the line L@; for the elements Dy-—U, and in the second column the sum of the frequencies of the La; and the Ma, line. The third column gives the absorption-edge L,; For W to U Duane and Patterson’s values are used, for Ta I have used my value of 8;, which must have nearly the same value as 1,, for Lu, Er, and Dy the values for L; have been interpolated. Of course, we cannot draw any certain conclusions from the last values, but still it seems to me very improbable that the frequencies of L; for Lu, Er, and Dy should be appreciably * Dolejsek, ‘ Nature,’ cix. p. 582 (1922). and the Theory of Atomic Structure. 563 larger than the values given in the table. The last columns give the differences :— A, =(La,+ Me,)— L@;, and A,= L, — (Le; + Ma). As may be seen from diagram [V., A, represents the energy difference between the O; and the N, level. For the ele- ments with lower atomic number than Pb the N, level appears to lie higher than the O; level, the difference between these levels increasing with decreasing atomic number. As one unit of the frequency, i.€., the Rydberg number, corresponds to 13°45 volt, we may ‘conclude that for Lu, ae and Dy the N, level lies about 45 volt higher than the Q; level. A, represents the binding energy of the 4, electron (see diagram IV.), The discontinuity appearing in the table for the value of A, for W is not essential. Using the author’s value of 8; for L, of this element, instead of Duane’s and Patterson’s value for Ly, we get i W : A,=2°42, a value which agrees fairly well with the values for the other elements standing in the table. It appears from the table thatit is most probable that for Dy the binding energy of the 4, electron at least must be Jess than 20 volt, thus being of about the same magnitude as the binding energy of the valency electrons. As to the binding energy of the 5, electrons, we have to consider two values: one corresponding to the QO; level and one corresponding to the O,level. As may be seen from the table, the first value is at least more than 45 volts for Dy, the second value is given approximately by the frequency difference of the L; absorption-edge and the line Ly,. From a discussion in Part III. § 8 it follows that the latter difference is likely to be more than 25 volts for Dy. From this we see that, where the 4, electrons appear for the first time in the periodic table, they are more loosely bound than the 5, electrons; for the elements with higher atomic number than Pb, however, the 4, electrons are definitely bound more firmly than the 5, electrons. § 3. It is of interest to consider other regions in the periodic system where we might expect to meet with pheno- mena analogous to those discussed in the last paragraph. In the neighbourhood of the iron group the M-shell develops from a shell containing four 3, electrons and four 3, elec- trons into a shell containing six 3), six 3,, and six 3, electrons. When the 33; electrons appear for the first ble some electrons of the N-shell are already present. Numerical 202 564 Mr. D. Coster on the Spectra of X-rays data about the binding of the 3; electrons in this region can only be obtained for Cu and Zn, for which elements the M-shell even has been completed. For this purpose we may compare the frequency of the line K®, with the sum of the frequencies of Ka, and Le. They are given in Table XXVI. TasLe XXVI. Frequencies. Kp. Ves pili: Cah a Oa 661°30 661:24 ZT 709°87 | 710°88 The values for the K-lines have been taken from the new measurements of Siegbahn and Dolejsek *, the Le, line has been determined by Hjalmar. This table shows that the binding energy of the 3, electron for these elements is not much different from that of tlfe electrons which are present in the N-shell. For Ag the N-shell has been completed for the first time into a shell of three subgroups of 6 electrons each and respectively with the quantum symbols 4), 4,, and 4,. From a comparison of the L,-absorption edge and the line LB, it appears that for this element the binding energy of the 4,-electron is.not more than 10 volts, thus being of about the same magnitude as the energy of the 5-electron which is already present. For the elements with higher atomic number, however, the binding energy of the 4,-electron increases more rapidly than that of the 5,-electron. In the same way it appears from a comparison of the frequencies of the line L@; and the absorption edge L, for Au, where according to Bohr. the O-shell even has been completed into a shell containing six 5,, six 5,, and six 5,-electrons, that for this element the d3-electrons are bound with nearly. the same energy as the 6,-electron. | In this connexion it may be pointed out that the rare- earth metals form a particular group of elements which in certain respects has no analogy in the periodic system. In fact, as the N-shell develops from a shell of three sub- ie each of six electrons, with the quantum symbols 4,, 4,, and 4, into a shell of four subgroups, each of eight electrons, with the quantum symbols 44, 45, 4,, and 44, the O-shell has already a certain completion, which was first * Siegbahn and Dolejsek, Zeitschrift fiir Physik, x. p. 159 (1922). and the Theory of Atomic Structure. 565 reached for the rare-gas Xe containing a subgroup of four 5, electrons and.a subgroup of four 5, electrons. Tor the other regions of the periodic system, however, where, accord- ing to Bohr, an inner shell of electrons is being completed (in the neighbourhood ofthe iron group, of the palladium group, and of the platinum group) no other shell with higher total quantum number than the shell in question has reached a stage of completion. §4. The new measurements of Siegbahn and Dolejsek in the K-series * show that for the elements with lower atomic number than 18 there seems to be some irregularity in the relation of the frequency of the line Kf, to the atomic number. This irregularity has been connected with the development of the M-shell. For the elements in the neigh- ‘bourhood of the iron group, however, where the second stage of this development takes place, no such irregularity was observed. It could perhaps be supposed that there should exist such irregularities for the lines L8,, Ly, Ly2, and Ly, in the region of the rare-earths, and that the appearance of the satellites on the long wave-length side of these lines should be connected with this fact. Extrapolating these lines, however, from the values for the elements which either precede or follow the rare-earths in the periodic table, we get in both cases values which agree fairly well with those of the measured lines, and at any rate agree much better with those than with the values for the satellites. If there are some irregularities in this region—as we should expect from what has been said about the binding energy of the 4,-electrons (Part III. § 8)—they must lie within the limits of experimental error. In this connexion, however, it would be of interest to measure the lines Bs, ¥1, y2, and y; for the rare-earth metals and for the elements in the neighbourhood of this group with the method of high pre- cision recently developed by Prof. Siegbahn. Unfortunately, this would be very expensive work. §5. As has already been stated in Part III. § 8, it may be assumed that the remarkable satellites, Qy4, yo, and yo; dealt with in Part IV. § 11, are connected with the comple- tion of the inner N-shell. A possible explanation of these lines is suggested in Part V. § 9. Here I shall only con- sider certain special points. Though the satellites on the short wave-length side of 8, and y, (2.e., By, and By, and ¥7) are very faint for the region Ba—Lu, and though they are partly overlapped by other lines, we may conclude, from a thorough examination of the plates, that there is not the * Siegbahn and Dolejsek, doc. cit. 566 Mr. D. Coster on the Spectra of X-rays slightest indication of anomalies in this region for these satellites. Therefore, we may assume that there exists no simple numerical relation between the satellites on the: long wave-length side and those on the short wave-length side of 8, and y¥,. It is difficult to say whether the line denoted as yo should be connected with y, or with y3. As yo gets closer to yo for elements with higher atomic number than Nd, and finally coincides with this line without crossing it, we are inclined to assume that yo is only connected with yp. We might expect to meet such anomalous satellites as those described in this paragraph in other regions of the periodic system where an inner shell of electrons is being completed. Thus, in the case of the elements in the region of the palladium group, we should expect anomalies for Liy2,3, LB,, and Ly, and for K®,. The authors who have investigated the K-lines in this region do not mention any anomaly for K@,. This line, however, is very faint. As to Lyp,3, it might be that the nes eet and 3631 for Rh and 7=3450 and 3433 for Pd are such anomalous satellites of Yo,3 (compare Part III. § 5); but they lie on the short wave- length side of this line. For the lines L@, and Ly, in this region no new satellites have been found up to the present. It should, however, be worth while investigating this region again with quartz * as analysing crystal, making use of the Imperial Eclipse photographic plates. Further, we might expect anomalous satellites in the region of ‘the iron group, where, according to Bohr, the M-shell develops from a shell containing four 3, electrons into a shell containing six 3), six 3,, and six 3, electrons. In this region Hjalmar { actually observed a satellite on the long wave-length side of K,, which he denoted as f’. Wentzel t has already suggested that the appearance of this line should be connected with the development of the M-shell. In their recent paper Siegbahn and Dolejsek observe that this line is much broader than the breadth of the slit of the spectograph, and that it was not possible to ‘separate this line from K;. From this and from the theoretical interpretation of this line (see Part V. § 9) we might conclude that it is hardly probable that this line is * As has been pointed out by Prof. Siegbahn, the grating constant of this crystal lies between those of gypsum and calcite, making it especially adapted for the region of wave-length in question. + Hjalmar, Phil. Mag. xli. p. 675 (1921). t Wentzel, Annalen “a. Pnysik, \xvi. p. 437 ee and the Theory of Atomic Structure. 567 identical with the line denoted by Hjalmar as @, for the elements Mg-S, as has been supposed by Sommerfeld *. Anomalies in the region of the platinum group, which we might expect for the lines LA;, Lys, Ly,, and perhaps also for the line L§;, have as yet not been established. Only the line L8; seems to be a double line in this region. As, how- ever, for the corresponding line Lf, in the region of the rare-earth metals no anomalies were observed, it is hardly probable that this structure of 8; should be connected with the development of the O-shell. § 6. As has been stated at the end of Part ITI., most of the non-diagram lines lie on the short wave-length side of an intense diagram line, their frequency difference with this line being approximately proportional to the atomic number. Moreover, it was suggested that these lines might be emitted by an atom which had lost more than one electron. In the meantime, a very interesting paper has been pub- lished by Mr. Wentzel f, in which he treats the non-diagram lines of the X-ray spectrum. In this paper Mr. Wentzel was able to show that the lines measured by Hjalmar on the short wave-length side of the Ka, line for the lightest ele- ments are emitted by atoms which have lost more than one electron, and he could account in a very suggestive way for the simple numerical relations which hold for the frequencies of these satellites, and for the order of magnitude of the frequency differences of the satellites and the diagram-lines with which they are connected. In the further elaboration of his theory, however, and especially in the part of his paper dealing with the fine structure of the absorption discontinuities, Mr. Wentzel has made some assumptions which seem to be rather unsatisfac- tory from a theoretical point of view t. In Part ILI. § 9 of this paper I suggested that under certain conditions the atom might lose more than one electron at the same time by an impact with one single high-speed §-particle, and that the regeneration of such an atom should be accompanied by the emission of a line, which should lie at the short wave-length side of an ordinary diagram-line. Mr. Wentzel, however, supposes that the electrons of double or threefold “ ionized ” atoms have been removed one by one, thus assuming that the atom, which has already lost one or more electrons, remains * Sommerfeld, Zeitschrift f. Physik, v. p. 1 (1921). + Annalen d. Physik, 1.c. : t Mr. Rosseland, who first called my attention to this fact, will soon give a theoretical discussion of the problem. —— = - 568 Mr. D. Coster on the Spectra of X-rays in this state long enough to be deprived of one electron more, before its regeneration takes place. § 7. As this point is of fundamental importance for the understanding of the laws which govern the constitution of the atom, I have tried to get some experimental information about this question. Though the experiments have only recently been started, the results hitherto obtained seem to be inconsistent with the assumption of the successive ioniza- tion of the inner shells of the atom. As has already been pointed out by Mr. Wentzel himself, from his theory we might expect that the intensity of the lines which are emitted by double ionized atoms should increase with the second _ power of the intensity of the corresponding ordinary diagram- lines. In Part II. § 7 the satellites of the lines La, and LA, have been discussed. It was shown that for Ag and the elements with lower atomic number, the satellites of a, have a rather complicated structure. This line seems to be accompanied on the short wave-length side by an emission-band and two rather sharp lines. With the new photographic plate I was able recently to establish a third very faint line of still shorter wave-length than the line denoted as a. After some preliminary examinations five photographs were taken of the Le lines of silver under the same condition as regards position of the crystal, but under the following different conditions :— | Tension. | Current. Time of exposure. | Plate Ro: Tove spark gap. h i9) m.a. 3 bt | qi FLT: 8000 volts. 40; 40 [iis teers 8000 _,, Oe 80- | ee Ws eae a 4700 ,, 20." 23 hours. | aN eae etki 9 mm. spark ee | Oise, 15 minutes. For the Plates II., III., and IV. the tension was read with a Braun electrometer giving the mean tension on the tube. No appreciable difference was found for the Plates I., IL., and III. This means that in the case of Plate II., the radia- tion being about 2 times as strong as in the case of Plate III., no variation in the relative intensity of the diagram-lines and and the Theory of Atomic Structure. 569 the satellites was observed*. It is very difficult to draw any definite conclusion from a comparison of Plate I. on the one side and Plates II., III. on the other side, as these plates have been taken with different tensions. But at any rate it is very remarkabie that no appreciable difference between these plates could be observed. A great difference, however, was observed between Plate IV. and the other plates. On this plate only one line could be observed on the short wave-length side of «,. This line was extremely faint, and seemed to have about the same wave-length as a. As the diagram-lines on this plate seemed not to be quite as strong as on the other plates, a fifth plate was taken under the same conditions as Plate I., but with a time of exposure of 13 minute. On this plate the lines a, and a were certainly not stronger than on Plate IV., whereas the satellites could very well be seen. The great difference between Plate IV. and the other plates is easily explained if we assume that by an impact with one ‘single @-particle more than one electron at the same time may be removed from the atom. The energy connected with the removing of an electron from the Ly-level corre-— sponds for Ag to 3350 volts. Thus, if two electrons should be removed at the same time from the L-shell, the tension on the tube must be at least more than 6700 volts. It is hardly probable that the maximum tension on the tube should have this value in the case of Plate 1V.; therefore the lines corresponding to a double ionization of the L-shell could not appear on this plate. The one satellite which still was present on Plate IV. may be due to atoms which have lost at the same time one electron from the L-shell and one electron from the M-shell. : Further, these experiments indicate that it should be possible not only to obtain information about the state of the atom in which a certain satellite is emitted, but also to gain more insight into the laws governing the disturbance of an atom by an impact with a @-particle. This was shown by some photographs taken of the fA, satellites (8), and 8,.) of silver with different tensions. These lines are very broad and diffuse. Under conditions differing about the same as those of Plates I. and III. in the above table, no change in * From a comparison of Plates II. and III. it appears that the intensity of the spectral lines is not proportional to the current through the tube. There is no real difficulty in this, as we have to expect that the intensity of the spectral lines is proportional te the current density in the focus- spot on the anticathode rather than to the total current. 570 Mr. D. Coster on the Spectra of X-rays the relative intensity of these lines in regard to the line 6, was observed. Photographed with the higher tension, how- ever, they seem to be much more intense on their long wave-length side than on the other side. This difference was not observed on the plates taken with the lower tension. I intend to continue these researches, using a direct- current source of high tension. 7 § 8. From these results and the considerations of the former paragraphs it is evident that we cannot agree with the theoretical interpretation given by Mr. Wentzel of the fine structure of the absorption discontinuities. It is well known that the researches by Fricke*, Stenstrom f, and Hertz t showed that at a short distance from the principal discontinuity another discontinuity may be found. The photographs give the impression that there are two white lines; the most intense of which usually lies on the long wave-length side. From this we may conclude that two different, rather definite, frequencies are selectively absorbed. According to Wentzel, the principal discontinuity should be connected with the removal of the first electron from the’ corresponding shell to the outside of the atom, the second with the removal of the second electron. As it seems very improbable that this successive ionization really takes place, it seems to be more likely that this structure of the absorp- tion-edge is connected with the conditions at the outer side of the atom, as has been suggested by Kossel §. In this connexion it may be pointed out that in this laboratory Mr. Lindh has found rather great differences for the wave- lengths of the principal discontinuities of the same element in different chemical compounds. § 9. For the elements Ti (22), V (23), and Cr (24) Fricke also found a second discontinuity lying on the long wave- length side of the principal discontinuity. Itis very probable that this anomalous discontinuity, as already has been sug- gested by Wentzel, is connected with the completion of the inner M-shell, which, according to Bohr, takes place in the neighbourhood of the iron group. I shall give an explana- tion of the appearance of this anomalous discontinuity, which differs from that given by Wentzel in some essential details. Wentzel suggests that in the region of the iron group the M-shell may occur in different modifications in different atoms of the same element, and especially that there are * Fricke, Phys. Rev. xvi. p. 202 (1920). + Stenstrom, Dissertation Lund, 1919. t Hertz, Zeitschrift f. Physik, iii. p. 19 (1920). § Kossel, Verhandlungen d. D. Phys. Ges. xviil. p. 339 (1916). and the Theory of Atomic Structure. o71 found some atoms for which the M-shell has one electron more than usual. The principal discontinuity should be connected with the removal of an electron from the K-shell of a “normal” atom, the anomalous discontinuity with the removal of an electron from the K-shell of an atom whose M-shell has one electron more than usual. I, however, suggest the following explanation. We will assume that the atoms of the same element have all the same initial state. ‘The removal of one electron from the K-shell by absorption of X-ray energy may, however, happen in different ways. The electron may be removed wholly to the outside of the atom, a process which should correspond to the principal discontinuity, or the electron may be removed from the K- and enter the M-shell, which for the elements in question is in a state of development*. The latter process should correspond to an anomalous absorption-line lying on the long wave-length side of the normal absorption-edge. Besides, we may assume that, if in the latter case the re- generation of the K-shell takes place by an electron of the M-shell, this should give rise to a line which should have exactly the same wave-length as the corresponding absorp- tion-line. Such a line has actually been found by Hjalmar ft (compare Part V. §5). If this explanation is correct, we should have in this line an example of a line which appears at the same time as emission and as absorption-line. A line of this type is only possible in a region of the periodic system where the initial level corresponds to a shell which is in a state of formation. Now we may assume that in the rare-earth group where the N-shell is in a state of formation we should have the same phenomena. [For these elements an electron may be removed from the L-shell under two different conditions : it may be removed wholly to the outside of the atom, or it may be transferred from the L-shell to the N-shell. The first transition should correspond to the normal absorption-edge, the second to an absorption-line lying on the short wave- length side of the edge. Besides, the inverse process of the second transition should give rise to an emission-line lying on the Jong wave-length side of a diagram-line, and having exactly the same wave-length as the corresponding absorp- tion-line. Anomalous satellites lying on the long wave-length side of diagram-lines have actually been found in the emission * Compare N. Bohr, Zeztschr. f. Physik, vi. p. 1(1922). See especially p- 60, where the possibility of such a transition has been suggested. t Hjalmar, Phil. Mag. xli. p.675 (1921). Hjalmar calls this line Kp’. 572 Spectra of X-rays and Theory of Atomic Structure. spectrum of the rare-earth metals (compare Part IV. § 11 and Part V. § 5). If these lines may be explained in the above way, we must expect that they also appear as absorp- tion-lines. In this case, however, we should have the remarkable fact that the difference between these absorption- lines and the corresponding normal absorption-edges should be quite considerable (é.¢., in the case of L, for Tb about 4() X.U. corresponding to 250 volts). I intend to start an experimental investigation about this matter. Summary. This paper has been divided into 5 Parts: Parts I., I1., and III. have already been published in Phil. Mag. xliii. p. 1070 (1922). In Part I. the general laws governing the emission of the characteristic X-ray spectrum have been dealt with and the relation between these laws and Bohr’s theory of atomic structure has been discussed. Parts II. and IV. contain the new experimental results, Part II. gives the new measurements in the L-series of the elements Rb-Ba, Part IV. those of the elements La—Lu. These measurements comprise: 1°, a great number of lines which may be arranged in a simple diagram, these are denoted as diagram lines; 2°, some non-diagram lines ; 3°, some absorption discontinuities. - Parts III. and V. give a theoretical discussion of the new measurements. From this discussion it appears that the new results as regards the diagram lines are in beautiful agreement with Bohr’s theory as regards the successive development of the shells of electrons in the atom (Part ILI. ney eee to Bohr, at different stages of the periodic table we meet with atoms for which an inner shell of electrons is completed. Thus the M-shell is completed in the neigh- bourhood of the iron group, the inner N-shell is partly completed in the region of the Pd group and again definitely for the rare-earth metals. The appearance of the Pt metals is connected with the partial completion of the inner O-shell. This conception is found to be in agreement with the experimental results. Especially for the completion of the N-shell in the region of the rare-earth metals several experimental proofs have been given (Part III. § 8). Again, according to Bohr, we must expect that, where an inner shell is. being completed, the most loosely bound elec- trons of this shell are bound not more firmly than the 9 Vibrational Responders under Compound Forcing. 573 valency electrons belonging to the outermost shell. From the experimental data of this paper it could be proved that this is really the case (Part V. $$ 2 and 3). In Part V. §§ 6-9 a theoretical discussion has been given of the non-diagram lines and of the fine structure of the absorption discontinuities. An experimental proof has been given that part of the non-diagram lines are emitted by atoms which have lost more than one electron at the same time by an impact with one single high-speed §-particle (Part V.§7). Furthermore, an explanation has been given of some non-diagram lines heh lie at the long wave-length side of diagram lines, and it has been suggested that these lines should | appear at ieee same time in the emission and in the absorptien spectrum (Part V. ¢ 9). I am much indebted to the kindness and interest of Prof. M. Siegbahn and Prof. N. Bohr. LI. Vibrational Responders under Compound Foreng. By Prof. EH. H. Barron, F.A.S., and H. M. Brownine, (SU a . [Plates III. & IV.] ‘a previous papers ¢, experiments were described in which sets of responders were under double forcing, but these forces were quite independent of each other. The present paper deals with cases in which the vibrational responders in use are under forcing, either (1) from the compound harmonic motion of a single-pendulum driver, or (2) from the associated motions of two coupled pendulums. In the first case they illustrate the analysis of a musical tone by the mechanism of the ear on the resonance theory of audition. In the second case they show the double resonance sometimes observed when dealing with two electrical circuits closely coupled, or two communicating resonating chambers used with thermophones. Compound Harmonie Vibration.—To illustrate by a set of responders the analysis of a compound harmonic vibration, the apparatus was arranged as shown in fig. 1. A stout cord ACB is fixed at A and B and set in motion by the swinging of the pendulum CD. The mass of the bob D is made paramount, in order that it may be used as the driver f. * Communicated by the Authors. + Phil. Mag. vol. xxxvii. pp. 453-455, April 1919, and vol. xxxviii. pp. 163-173, July 1919. { From this it follows that ABCD keeps the shape shown during vibration of D; hence the virtual length of this driving pendulum is ED. > — — rn d74 Prof, Barton and Dr. Browning on The lengths of the twenty-five responders (RI....I7...III..8) range from one to sixteen, and are in geometrical progression. The distances from A of their suspension points are pro- portional to their lengths, so as to ensure that each shall receive an equal forcing from the driver. The advantage of this geometrical progression of length lies in the fact that the “intervals” (to use the musical expression) between adjacent responders are then all equal. Indeed, the whole set of twenty-five responders forms two octaves of the Fig. 1.—Apparatus for Harmonic Analysis. es 4 { Ra chromatic scale. It is to be noticed that these light vibra- tional responders have virtual lengths from their bobs to the cord AC and not to AEB. Their bobs consist of paper cones, with the addition of a ring of copper to diminish the damping to a suitable extent, and so make the resonance or response just sharp enough. The bobswere made precisely alike, and this ensured the damping being the same for each responder. When first the bob D was allowed to swing freely, only one resonance maximum occurred in the vibration of the responders at the: part marked I1 in the diagram, thus showing that the motion was simple harmonic of the frequency thus indicated. Next, the bob D was moved by hand in a compound harmonic motion of the same fundamental frequency as was natural to it. Then the responders showed the fundamental Vibrational Responders under Compound Forcing. 575 frequency as before, but showed also by higher maxima the quicker vibrations which are the harmonic components of the compound motions now executed by D. Three cases of such harmonic analysis were carried out with this apparatus and the vibrations and results are shown in figs. 1, 2, and 3 of Pl. III. The curves shown below each indicate the motions given to the bob by hand, and the photographic reproductions above give time-exposures of the responders. In fig. 1 (Pl, III.) the motion is compounded of vibrations of relative frequencies one and two, or tone and octave, to adopt musical language. In fig. 2 the motion of the bob was com- pounded of vibrations of frequencies one and three, and the result of analysis is seen to be that of a tone and its twelfth. In fig. 3 the motion of the bob, as shown below, is com- pounded of frequencies one, two, and three, and the responders give maxima at the corresponding places, which the musician would call tone, octave, and twelfth. Coupled Vibrations.—The arrangement now adopted is shown by the reproduction in fig. 4, Pl. III. There it may be seen that the two pendulums of nearly equal mass (each suspended by a bridle and vertical cord) are coupled by the bridge across the near part of the bridle. On the far part of the bridle of one pendulum a set of thirteen responders are in use. ‘These are of precisely the same type and arrangement as the twenty-five used for the harmonic vibrations, but here only one octave of the chromatic scale is provided instead of two octaves. (The white cones which constitute the bobs are clearly visible, but the black suspension threads do not show.) For convenience, the responders may be uamed according to the chromatic scale of C. It was recently found to be far more convenient to have the bridge adjustable along the bridle instead of fixed at the junction of the bridle and vertical suspension for the heavy bob. If a line is drawn from the bridge across either of the coupled pendulums with heavy bobs to the distant end of the bridle, and the suspension cord imagined to stretch vertically upwards to meet this line, then the completed length of the suspension gives the vibrating length of the coupled pendulum when the bridge is held at rest, and therefore this length defines the quick period peculiar to the coupled system with the coupling in question. Fig. 5 shows the effect of starting the vibration by burning a thread which held the bobs near together. In this case the bridge (as shown in the photograph) remains at rest, and the bobs execute the quick vibration alone. This is exhibited by the responders, which show a maximum amplitude at a 976 Vibrational Responders under Compound Forcing. frequency between those of the fifth and sixth responders counting from the shortest (or in musical terms between Ap and 6). In fig. 6, on the contrary, the effect is due to starting the pendulums simultaneously and in the same phase, so that each swings with bridle and suspension remaining in a plane. This isolates the slow motion of which the coupled pendulums are capable, and the responders now show a maximum ampli- tude at a frequency between D and DD), to use the musical terms. The longest responder, C, does not show in the photograph. In Pl. IV. are seen the effects of starting one pendulum while the other hangs free. This results in the quick and slow vibrations being performed simultaneously by each pendulum. Fig. 7 begins with about 10 per cent. coupling. This leads to the execution of frequencies so near alike that it is difficult to discriminate between them in the phetograph. The response is here seen to be spread upwards as compared with fig. 6. In actually watching the responders, the beats helaeen the two rates of forcing were clearly visible. For a coupling of tifteen per cent., as shown in fig. 8, two maxima are. distinctly visible. Musically speaking, the notes D and EH are responding best, and the interval between them is one tone. In passing from fig. 8 to 9, fig. 10 to 11, and fig. 11 to 12, the position of the upper maxima rises by one responder at a time, and it may be said musically that the pitch is raised by a semitone each time. Between figs. 9 and 10 the pitch rises by a tone. Thus in fig. 12 the responders called D and A will be found to be the maxima, and the musical interval to be a perfect fitth. The couplings required for the various responses are shown on the Plate for each figure. In order to show that the responders are responding accurately to the two vibrations of the coupled pendulums, traces might be taken on a board moving perpendicularly to the pendulums. But as this has been done for the pendulums alone, comparisons can be made between the figures on Pl. IV. of the present paper and Plate V. of ‘ Vibrations under Variable Couplings,” Phil. Mag. vol. xxxiv. Oct. 1917. It will be seen that fig. 12 of the present paper and fig. 7. of the Oct. 1917 paper have the same coupling, and that the ratio of the ad: of the two component vibrations is approximately 3: 2 in each case, though exhibited in entirely different ways. Nottingham, May 31, 1922. yhSamy J LI. The Measurement of Light. By NorMAN CAMPBELL, Sc.J)., and BERNARD P. DupDING”. y ) ; Summary. T is maintained that in order to establish a scientifically or legally satisfactory system of measuring any physical magnitude it is not sufficient to define the units to be em- ployed; it is necessary also to state the laws of measurement involved. Photometry provides an exceptionally favourable illustration of this necessity ; the laws of measurement underlying photometry are therefore considered in’ some detail. 1. International congresses have from time to time fixed with great elaboration the units of certain practically impor- tant physical magnitudes. The results of their labours are embodied, not only in scientific treatises, but also in much national legislation. It appears to be thought widely that, when the units have been fixed, an entire system of measure- ment has been established and that no further question can arise concerning the value to be attributed to any magnitude. Weare of the contrary opinion. It appears to us, for example, that when the unit of resistance is fixed, we know certainly that one definite body has a resistance of 1 ohm ; but we do not know certainly what other bodies have a resistance of 1 ohm or what bodies, if any, have a resistance of 2 ohms. And knowledge on these matters, which are not decided, is . of much greater importance from every point of view than knowledge of those which are decided. Let us put the matter practically. An Act of Parliament _ has laid down what is one ampere and one volt, and decided that one watt-second is the energy required to maintain one ampere under a potential of one volt for one second. Well and good. But when we are presented with our electric-light bills, we are asked to pay for a good many kilowatt-hours. . And the Act has never told us what kind of current, under what kind of potential, for what kind of time, is the precise number of kilowatt-hours for which we are asked to pay. It appears to us that the Courts ought to decide that we cannot be legally forced to pay for anything but one watt-second, because otherwise we are paying for something of which the nature is wholly unknown to the law. Doubtless counsel would advise us that, if we tried to * Communicated by the Authors. Phil. Mag. S. 6. Vol. 44. No. 261. Sept. 1922. 2.7 578 Dr. N. Campbell and Mr. B. P. Dudding on maintain such a position, the judicial mind might suffer one of its occasional lapses into common-sense. But while we would accept his view of the probable results of the action, we would protest that common-sense has nothing to do with the matter. It is not common-sense that tells us how to measure watts; it is the wisdom of the giants of physical science accumulated for more than a century. If we know quite definitely how to measure watts, we know all the fundamentals of electrical engineering and much of experi- mental physics. All measurement depends upon laws, and these laws usually include most of the important experimental facts of the science concerned with the magnitudes in ques- tion. Knowledge of these laws is not common property, and can no more be assumed than a knowledge of the customary units. | In opposition to our contention that a statement of the laws of measurement should be included with a description of the units in any defining legislation (by which we refer now to the decisions of scientific committees rather than to those of national legislatures), two arguments may be used. First, it may be said that the choice of units, being arbitrary, is suitable for legal decision, while the truth of laws, being wholly free from arbitrariness, is not; the first, therefore, but not the second, is suitable for inclusion in legislation. But surely, even in legislation, relevance is of some import- ance. Our only objection to present legislation on such matters is that it is totally irrelevant to all cases of practical importance ; all that we suggest is that something should be added which will make it relevant. Secondly, it might be said that to meet our suggestion would be to convert the legislation into a complete treatise on science. But such a treatise would be either too large or too small for the purpose. It would be too small if we were urging that the impossible task should be performed of stating all the experimental laws that are concerned ; it would be too large if we are urging (as we do) that the laws directly concerned should be indi-. cated with a reasonable amount of precision. If we are asked what is meant by “directly.” and “ reasonable,” we can only reply by giving an example. And that is what we propose to do. 2. Photometry seems a suitable system of measurement to choose; its measurements (even so far as units are concerned) have not yet been fixed completely with international con- ‘sent; the laws of measurement concerned present some features of intrinsic interest; and some of the attempts at establishing systems of measurement that have been made the Measurement of Light. d79 appear to us exceptionally unfortunate. Thus the inverse- square law, upon which most practical photometry depends, is treated very inadequately. The units cannot possibly be defined without reference to the law. It is usually implied, and not openly stated, as we hold it should be: and state- ment by implication is peculiarly objectionable here, because the law is only true in very special conditions which cannot be sufficiently described. But it is more serious that it is treated as a primary and necessary law for the measurement of all photometric magnitudes, so that the only form in which it can be stated with full precision is that photometric measurement is possible. This view seems to us utterly mistaken ; one photometric magnitude at least can be defined without reference to it; and when that magnitude is de- fined the proposition can be stated as an independent law concerning it, and can be subjected to direct experimental proof. Closely connected with this defect is the total absence of any clear statement of the meaning of the addition of illumination, although in the practical use of light that law is much more important than the inverse-square law. It is more important to know that we can double generally the illumination by doubling the number of lamps than that we ean multiply it four times by halving their distance. Lastly, the significance of the magnitudes is made to depend wholly upon a theory of illumination—the theory which may be baldly stated in the terms that illumination is due to the incidence of something called light. That theory is wholly unnecessary; and since, like all theories, it might be doubted, it should be studiously avoided and all definitions framed in terms of experimental concepts only. With these preliminary remarks we will proceed to busi- ness. Perhaps the reader should be warned that what follows is not put forward as a model of an actual formulation, but merely as a sketch of the principles to be followed. ‘The actual formulation would require much greater verbal pre- cision and much less explanation of reasons for the scheme adopted. The nomenclature of concepts concerned with measure- ment adopted here is that employed by one of us in ‘ Physics,’ Part II. 3. Light-measurements are based on judgments of equality of brightness of photometric surfaces. These judgments are made by the direct perception of normal persons— normality in this matter meaning simply agreement with the great majority of mankind. a gM ad nd 580 Dr. N. Campbell and Mr. B. P. Dudding on Photometric surfaces (P.S.) are members of pairs. A pair of photometric surfaces is defined by the condition that, if their positions are interchanged while everything else remains unaltered, equality of brightness is undisturbed whatever the nature of the illumination. P.S. which form a pair with the same P.S. form a pair with each other. The condition laid down ensures that the surfaces shall be of the same shape, including radius of curvature, of the same colour, of the same reflexion and diffusing coefficients, and of the same intrinsic luminosity. Ifit is fulfilled, it would be possible to use the pair of P.S. for light-measurement—unless, perhaps, if they were absolutely black—for all or for some colours. But it is convenient to choose surfaces which fulfil as nearly as possible other conditions which ensure that the surfaces are white, matt, and non-luminous. White is here used to include grey. A white surface is one such that light reflected from it is of the same colour as the direct light whatever the nature of that light; similarity of colour is judged by direct perception. A pair of matt surfaces are such that, if they are equally bright when viewed from one direction, they are equally bright if viewed from any other. No actual surfaces fulfil these conditions perfectly ; none are perfectly white or perfectly matt. But some surfaces fulfil the condition very nearly, and these are best adopted for photometry. Further, it is desirable to choose surfaces which are white (in the ordinary sense) rather than grey ; that is to say, if B, sub- stituted for A, is less bright than A, A should be preferred to B. And, lastly, it is desirable to choose surfaces free from intrinsic luminosity, that is, such as can be made to appear perfectly dark by placing suitable screens round them. dy 4. The fundamental photometric magnitude is ilumina- tion. A pair of perfectly matt P.S. have equal illumination (or are equally illuminated) when they are equally bright. But since, as we have remarked, there are no perfectly matt surfaces, the law of equality, interpreted according to this definition, is not strictly true; the P.S. may be equally bright if viewed from one position, but unequally bright if viewed from another. If, however, a considerable change in the directions from which the surfaces are viewed does not change their relative brightness, the law of equality will be true for observations made within this range, and the definition so far will be satisfactory. The question remains, however, what is a considerable range for this purpose ; it can only be answered perfectly definitely when the further definitions of measurement have been added ; it can then be | the Measurement of Light. 581 stated that the range must be such that the entire system of measurement established is satisfactory. But meantime conditions can be prescribed more nearly in which the defi- nition will be satisfactory even if the P.S. are not perfectly matt. There are two such conditions to be fulfilled: (1) The lines joining the two P.S. to the observing eye must make equal angles with the normals to those surfaces. (2) The lines joining the illuminating source (or sources) to the P.S. must make equal angles with the normals to those surfaces and with the lines joining the surfaces to the eye. The assumption that these angles are unique implies that the surfaces and sources are “points.” Again, it is impossible to prescribe exactly what are points, except by reference to the completed system of measurement that we are about to establish. But the proposition which is important for our purpose is that, if the dimensions of the surfaces and sources are made sufiiciently small compared with the dis- tance between them, the definition .applied subject to the conditions just stated will prove satisfactory, even if actual surfaces, not perfectly matt, are employed as P.S. We can therefore give instructions whereby satisfactory measure- ments can be made; and when once such measurements have been made, the range of permissible variation of the conditions is determined by the agreement of other measure- ments with those made under these standard conditions. Difficulties such as these in describing the precise condi- tions for satisfactory measurement occur in all branches of physics; but they are probably more acute in photometry than in most other branches. It will be observed that the absence of perfectly matt surfaces has forced us to introduce at this stage the ideas of ““a source” and illumination by a sonrce, which properly belong to a later stage. We may, therefore, describe them rather more closely. A surface is said to be illuminated by a source when its brightness can be changed by changing the physical condition of some body (e.9., by changing the current through an incandescent lamp) or by interposing suitable media between that body and the surface. The body, the condition of which thus affects the brightness of the surface, is a source illuminating it. “ We can now state the first of the important laws of illumi- nation which are not necessary laws of measurement of that magnitude. If a pair of equally illuminated P.S. are re- placed in position by another pair of P.S., the second pair will also be equally illuminated, even if a member of the 582 Dr. N. Campbell and Mr. B, P. Dudding on first pair does not form a photometric pair with a member of the second. Since any surface (except an absolutely black one) can be used as a P.S. with a suitable pair, this law naturally leads us to regard illumination as something characteristic, not of the surface illuminated, but of the circumstances in which it is placed. We find, further, that among the most important of these circumstances are the positions of the surfaces relative to the sources and the nature of these sources. It is these laws which give us the first clues to a theory of illumination ; but such a theory, though it is a very useful guide in seeking a satisfactory system of measurement, is best excluded entirely from any deserip- tion of it. 5. We must now define Pee The illumination on a surface X from the sources A and B is equal to the sum of the illumination of X from A and the illumination of X from B, if A and B, when they are illuminating the surface, are always in the same physical state and in the same position relative to the surface whether they are acting together or singly. With this definition, the first law of addition is true ; cutting off the illumination from a source always decreases brightness. But the second law is not true in all cireum- stances ; it is not true, for example, when the Purkinje effect is apparent. For, if R, and R, are red sources, B, and B, blue sources, and if the illumination from R, is equal to that from B, and that from R, to that from Bg, the illumina- tion from R, and R, will not be always equal to that from B, and B,. On the other hand, if all the sources are of the same colour, the second law is true; and it is true apparently if the sources, though of different colour, give sufficiently great illumination and fulfil some other minor conditions. The Purkinje and allied effects are simply failures of the neces- sary laws of photometric measurement, and any complete statement concerning such measurement must include the proviso that the conditions are such that these effects do not enter and that the laws of measurement are true. It must be insisted that the only logical way to describe these effects is in terms of the failures of the laws of measurement which they involve; to describe them in terms of the measurements which they make impossible, though it may be convenient and conduce to brevity for general purposes, is utterly ludicrous if precision is important. 6. We have now defined completely the magnitude illu- mination, and can proceed to measure it and to state signi- ficantly and prove experimentally the following important the Measurement of Light. 583 numerical law concerning it :—-The illumination on a surface A froma source X is inversely proportional to the square of the distance of X from A, so long as (1) X and A are “points? ; (2) the angles between the line XA and any lines characteristic of X and A remain the same; (8) the medium between X and A (or rather the variable part of it) is perfectly uniform and transparent ; (4) that the bodies sur- rounding X and A are perfectly black. Of course, as usual, these four conditions are really in part definitions of points, perfectly transparent media and perfectly black bodies, the significant proposition being that there are conditions in which the numerical law is true and that they are indicated by the crude meaning.attributed to the terms used. Though the principles involved in the fundamental mea- surement of illumination by means of the definitions of equality and addition are the same as those involved in any other fundamental measurement, it may be well to describe in some detail one form of experiment by which the inverse- square law might be proved. Two plane P.S. A and B are taken. A is viewed along its normal, and any sources X (all of which must be points) illuminating it are disposed in fixed positions on the surface of a circular cone of any apical angle with this normal as axis. B is viewed at any convenient angle, and the sources Y illuminating it are placed in any convenient positions with regard to it. Some one constant source Y, is chosen arbi- trarily and placed arbitrarily in some constant position relative to B. The illumination of B by Yo is arbitrarily chosen as unit. We proceed, then, to find two sources, X, and X,' such that either of them, acting alone, makes the illumination of A equal to that of B. Hach of these sources then gives unit illumination on A. X, and X,'are then made to illuminate A together; Y, is extinguished and some other source Y,is found which, placed in a certain position, makes the illumination of B equal to that of A. X, and X,' are now extinguished, and a source X, found which, in some position on the cone, makes the illumination of A equal to that of B. The illumination of A by X, is then 2. And so on for the other positive integral values of the illumination of A. Experiment shows that it is impossible to find sources which give negative values for the illumination. But frac- tional illumination can be obtained. In order to make the members of the standard series of illumination the value of which is 1/n, we have to find n sources placed on the cone such that any one of them illuminating A makes the illumi- nation of A equal to that of B when illuminated by some 584 Dr. N. Campbell and Mr. B. P. Dudding on source (to be discovered by trial and error) while all n of them acting at the same time make the illumination of A equal to unit illumination. And so on for the values m/n, and the completion of the standard series. We now illuminate B by some point source which can be moved along a straight line passing through B. When the distance of Y from B is 7, we find what member of the standard series gives the same illumination on A as Y on B. We then multiply 7? by the value assigned to this member, and find, after a large number of trials, that there is some value of this product such that we cannot find any law to predict whether any value of it resulting from single obser- vation will be greater than or less than this value. We have then proved the numerical law. 7. This numerical Jaw, like almost any other enables us to define a derived magnitude, namely, the constant product. I7r?. Hxperiment shows that this constant is indeed a magnitude, an ordered propertv of the system under con- sideration, determined by and in general variable with (1) the nature of the point source, (2) the constant imper- fectly transparent medium (if any) intervening between it and the P.S., (3) the direction of the line joining the source to the P.S. relative to lines characteristic of the source, (4) the angle 6 which that line makes with the normal to the P.S. We shall suppose that (2) is not important because all the medium is perfectly transparent. Hxperiment shows that Ir? is approximately proportional to cos 8. We might therefore define a new derived magnitude I7? sec 0, which would depend only on the source and on the direction of the P.S. relative to it: we might term it the intensity of the light emitted by the source in the direction of the P.S. and denote it by ®. But since the cos @ law is not accurately true, ® would not be truly a magnitude ; it is better to define ® as the value of Ir? when cos 0=1 ; it is then totally indepen- dent of the cos 0 law, and the limitation to.one value of @ is not practically tr oublesome. 8. ® is a function, not only of the nature of the source, but also of the direction of the P.S. relative to it.. We can eliminate the direction and obtain a magnitude depending only on the nature of the source, if we can form the sum \Bdo, where wo is the solid angle subtended at the source by the P.S.‘in any direction. The formation of this sum will be legitimate if the Inverse-square law is obeyed whatever the direction, so that there is a ® in every direction, and if ® is independent of w, the direction being the same; these the Measurement of Light. 585 conditions are fulfilled if the P.S. and the source are suffi- ciently small. JPdo when the sum is taken over all directions from the source is called the flux of light from the source ; if it is taken over a limited range of directions, it is called the flux emitted within the limit of those directions (fF). The name that we (and everyone else) have chosen for this magnitude is, of course, suggested by a theory of illumina- tion; but it is essential to notice that it can be defined wholly independently of any theory. The magnitude is very important, because it is closely connected with the energy lost by the source and, through that energy, with the magnitudes of other branches of physics. Accordingly, it is often useful to invert the relations at which we have arrived and to express intensity and illumination in terms of flux. It is apparent that intensity is the flux per unit solid angle subtended by the P.S. at the source, and illumination the flux per unit area of the P.S.; but it must always be remembered that these definitions are inverted and _ that really we know nothing about flux till we have measured illumination. Again, once we have arrived at the connexion between flux and illumination we may use this connexion to measure flux when the conditions which we have hitherto supposed necessary for its measurements are not fulfilled. Thus, even when a surface is illuminated by a source or sources which are not a single point obeying the inverse- square law, we may say that the flux incident upon it is the product of the illumination and the area. If we make this purely verbal definition of flux, we then find (in virtue of the law of addition of illumination) that the flux from many point-sources incident on a surface is the sum of that inci- dent from each of them ; when we can use a definition to state a law, the definition ceases to be purely verbal and becomes an expression of fact ; it may be admitted to our scheme on a parity with the other definitions of measurement. 9. One important photometric magnitude remains for discussion, brightness. We have used the conception of brightness before as something directly perceived, but we have not framed any scheme for measuring it. The magni- tudes already established enable us to measure brightness as a derived magnitude. Let us take several surfaces which are unequally bright, though each of them is uniformly bright, and allow them in turn to illuminate a P.S., and measure the illumination of that P.S. (from which we deduce the intensity ® of the light falling on it), the area S of the bright surface and the angle @ which the line joining 586 Dr. N. Campbell and Mr. B. P. Dudding on the P.S. to the surface makes with its normal. Then we find as a matter of experimental tact that, when the inverse- square law is true, ® is proportional to 8; and that if we hee , the order of this expression for S cos @ the different sources is the order of their brightnesses directly perceived ; the source for which the expression is the greater is always the brighter. That is equivalent to saying that this expression is a measure of the brightness ; and accordingly we define the brightness of the surface, now a magnitude, as the intensity of the light emitted by it in a given direction divided by the area of the projection of the surface on a plane normal to the direction of viewing. In general brightness so defined is a function of the angle «, in accordance with the fact that surfaces in general alter in brightness when the direction of viewing is changed. But there are certain surfaces, which are those most nearly matt, the brightness of which does not depend greatly on that angle. For such surfaces, the intensity of the light emitted at angle @ is proportional to cos a; 7.e¢., Lambert’s law is true. When Lambert’s law is not true, if brightness is to be defined uniquely, some convention concerning the relevant values of « must be introduced. It is usual* to define brightness only for «=0; another course, in some ways more satisfactory, would be to take the mean brightness over the whole range of a. So far we have regarded the measurement of the intensity of the light emitted by a known area of a bright surface as a mere means of measuring the brightness directly perceived. But for some purposes this intensity is important on its own account, e.g., when we are considering the illumination that the bright surface would produce. For the purpose of this illumination it is immaterial whether the bright surface is or is not uniformly bright ; so long as the intensity of the light emitted by unit area is the same, it does not matter whether the light comes from all parts of the surface or only from a few specks on it. Accordingly, it has become customary (though we think the custom unfortunate) to speak of all surfaces as equally bright so long as the intensity of the light from them per unit area (or possibly unit projected area) is the same, regardless of the fact that the apparent brightness of the surfaces, the quality directly perceived, is utterly different. 10. There remain to be considered certain subsidiary laws, form the expression * Cf. Winkelman, Handbuch der Physik, Optik, p. 747 (2nd Ed.). the Measurement of Light. ENE which, though not absolutely necessary to photometry, are useful in its more elaborate developments. It is doubtful whether they should be included in any official statement concerning photometric measurement ; for it is difficult to describe precisely the circumstances in which they are accurately true, and in the last resort measurements should always be checked by the basic methods that have been described so far. But three of them may be noted. The first arises when reflecting or refracting surfaces are placed hetween the P.S. and the source or (more often) the eye. We have then to note that the line joining the P.S. to the source or eye is to be taken as the optical path between the two, and all statements concerning direction or distance interpreted accordingly. The second arises:when a sphere- photometer is used to measure average flux or average candle-power. The jaw involved is then that the iUlumina- tion of the P.S. used in the measurement is determined wholly by the average intensity of the source in the sphere. This proposition is never true universally, and only experiment with each photometer can tell within what limits it is true. The last is Talbot’s law, employed in rotating sector methods or when asymmetrical sources are rotated to obtain average intensity. This law is apparently accurately true, and might therefore be included on an equality with the other laws of photometry. 11. Having stated the facts, we may proceed to conven- tions. We have to define the units of the various magnitudes. Since we have one fundamental magnitude (illumination) and three derived magnitudes (intensity, flux, brightness), we shall need n units arbitrarily assigned to some specified physical systems or substances and 4—n formal constants for the three numerical laws defining the derived magnitudes (n<4). We also need units of the other fundamental (or quasi- fundamental) magnitudes involved in the laws of derivation, namely, distance, solid angle,and surface. For many reasons it is convenient tomaken=1. But it is important to observe that it is not necessary to assign to the arbitrarily selected system unit value of the fundamental magnitude; we can assign to it unit value of any of the connected magnitudes and define the unit value of the fundamental magnitude as if it were measured as quasi-derived. This procedure is actually the most convenient ; we assign arbitrarily a value of intensity and not of illumination. We describe a physical system which we call a standard candle. We then assert that the intensity of the light emitted by it 588 Dr. N. Campbell and Mr. B. P. Dudding on in some direction, specified by its relation’ to lines charac- teristic of that system, is 1. (Intensity of standard candle, or 1 candle-power.) We choose the centimetre as unit of distance, square centimetre of area, and steradian of solid angle. We have now to choose the formal constants «, 8, y in the laws o=21y7) =B\ Pde, @D Hanne ane We have hitherto treated these constants as all being unity, but there is no reason, if we see fit, why we should not assign to them other values; and the choice of unity is as arbitrary as any other. Asa matter of fact, «, 8, y are all usually chosen to be 1, and there is no objection to that practice. The unit of illumination (phot) is then that of a P.S. distant 1 cm. from the (point) standard candle in the prescribed direction ; the unit of flux (lumen) that emitted by the standard candle within unit solid angle surrounding the prescribed direction, if the condition is fulfilled that the intensity is the same within all parts of that solid angle. The unit of brightness (lambert) is that of a perfectly matt surface which emits light of unit intensity when its area projected on the plane normal to the direction of viewing is one square centimetre. It is to be noted that these derived units (unlike the fundamental unit, which is not that of the fundamental magnitude) cannot all be actually realized. But such a feature of derived units is quite usual ; e.g., unit Young’s modulus cannot be realized. 12. It may possibly be useful to compare our nomencla- ture and notation with that adopted by some official body. We choose for this purpose that of the Standards Committee of the Optical a of America, described by its Chair- man, P. R. Nutting * Our Notation. American. Aiioinmatvon | oye) Flux density. .;.. D Intensity... p Iimpensity, os be C ID vox ae sag eva Cat F Lite SPOR ais Ba. F Brightness —...... B Brie nimess ee B Of course, notation is a matter of mere convenience ; but we have departed from the American scheme because we * Journ. of the Opt. Soc. of America, iv. p. 230, July 1920. the Measurement of Light. 089 think it is most highly inconvenient to denote the trul fundamental magnitude by a name (flux density) so highly suggestive of a derived magnitude. And we have preferred ® to C in order to indicate the close connexion between intensity and flux. But we have more serious differences with the American Committee. On p. 232 a lambert is defined as the bright- ness of a substance emitting one lumen per square centimetre of projected area in the direction considered. But to speak of a lumen emitted in a direction is to talk nonsense; the flux emitted within an infinitesimal angle is always infini- tesimal. In place of ‘Sone lumen” should be substituted ‘one lumen per steradian.” Since on p. 231 brightness is defined as flux per steradian per square centimetre, it is probable that the omission of the words ‘per steradian ” is a mere oversight—the more unfortunate because it is twice repeated. But the statement, also on p, 232, that “one candle per square centimetre equals 3°1416 lamberts” is less easily comprehensible. In the first place, lamberts have only been defined for surfaces which obey Lambert’s law ; for with other surfaces the lumens per steradian per unit of projected area will vary with the direction considered. Since a surface made up of “‘ candles per square centimetre” does not obey that law, it is meaningless to attribute to it any number of lamberts. To remove this objection two alterations can be made :—(1) For ‘candles’ can be substituted ‘ candle- power’; for candle-power is often used as a synonym for what we and the American committee call intensity *. (2) Some specification can be added concerning the direction in which the candle-power or intensity is to be measured. For (2) it is most natural to take the direction normal to the surface ; but if we take this direction, a surface of one square centimetre emitting an intensity of one candle-power (2. e., one lumen per steradian) has a brightness of 1 lambert, not a lamberts as the American committee affirm. We might also take average intensity over a hemisphere based on the surface. If the surface is plane and obeys Lambert’s law the average intensity over this hemisphere is 4 candle-power per square centimetre, and a surface emitting 1 mean candle-power per square centimetre would be equal to * E. g., average candle-power or horizontal candle-power means average or horizontal intensity. The term ‘‘candles”’ is often here, too, used in place of candle-power. We believe that this confusion between a thing and its properties is one of the main sources of the obscurity of photometric definitions. 590 Mr. D. L. Hammick on Latent Hea 2 lamberts. Doubtless some function of the intensity and the angle could be devised such that, in a corresponding sense, one candle-power per square centimetre would be equal to w lamberts, but we cannot discover a function which has the least intrinsic plausibility. Doubtless the American committee have succeeded where we have failed, but they have been unduly modest in concealing their ingenuity ; they may fairly be asked to explain how they have arrived at their surprising result and why they have preferred it to the simple and obvious convention which gives the result that 1 candle-power per square centimetre equals 1 lambert. May 27, 1922. LI. Latent Heats of Vaporization and Expansion. 3 By D. L. Hamoick*. A CCORDING to van der Waals, the pressure in a homogeneous fluid system is p so, When the ‘system expands, work is done against the pressure. The value of this work for the expansion of 1 gram of a liquid (volume x c¢.c.) until its volume is wv? c.c., the specific volume of the liquid when completely vaporized, is given by if | V2 LC bB) a = pdv+ { aye UV e ¥1 al Ney nee oe _ SSS, flict Ohta aie = 8 Bia a te When v, is large when compared with v1, this expression becomes a Jak — ae UT ais Uy \} or, putting pr, equal approximately to Mw REY a a a hac ae xe . ° ° ° ° (1) * Communicated by the Author. of Vaporization and Expansion. aon From (1) we have, at constant volume, dA 1nd Ne Beemer 4y fe LEAs ag a Te? ir On v Moin ag) Es : elated eae aT (= can) aah h Hence, substituting in the equation & dA A-U= dee > E dA) a wee Hoe — = a id pata >= — — () e e . . 2 8 A+ Tom pike al) (2) = Nex, the latent heat of the expansion. Now Davies (Phil. Mag. [6] xxiv. p. 421) has obtained the following expression for 7, the initial pressure in a fluid:— aes = (2t,—T) (v, and T, are eal volume and temperature). il Aan ae 1 peal Davies has also shown (Phil. Mag. [6] xxiii. p. 415) that 1 2 e . . Sav = 4, the coefficient of cubical expansion, so that 2T;=T 1 (da - a = =—=— —— O. 6 é . * . (3) But, if += = we get from (3): Lda a (Gr),= a her, = —(1 +a). Seer sn (a) ay Hence 1 (32) 1 Substituting in (2) When T=0, we find Nex = <3 in other words, at the absolute zero the expression becomes identical with Bakker’s expression for the latent heat of vaporization. ——————ESE i 592 Mr. D. L. Hammick on Latent Heats The question now arises as to the connexion between the value for the latent heat of expansion as given by (4) and the latent heat of vaporization at ordinary temperatures. The two latent heats would be equal, provided that no change in the internal energy of the substance occurred during the transition, at constant volume, from the liquid to the vapour phase. In other words, the condition for equality is: { cat ( cat =0. ey e/ I (C,, ¢ are the specific heats at constant volume respectively.) If (5) does not hold, then the difference between the latent heat of vaporization and Xex, of equation (4) will be oT { © 2 jar a Again, if during the passage from the liquid to the gaseous state a change in molecular aggregation occurs, a further quantity of heat, h, representing heat of asso- ciation, must be taken into consideration. Hence we aed Me = Me PHO In the Table values of > x, calculated according to equation (4) at the boiling-point, are compared with the observed values of Aya, the latent heat of vaporization. Values of “a” are given as atmospheres pressure+ (volume of 1 gramme molecule of gas at N.T.P.)? (Guye and Frederich, Arch, Sci. phys. et natur. Geneve, ix. p. 22, 1900). Specific volumes and a, the coefficients of expansion, are taken from Young (Sci. Proc. Roy. Dubl. Soc. xii. p. 414, 1910) and Tyrer (Trans. Chem. Soc. 1914, p. 2534). The latent heats of vaporization refer to 1 gramme of liquid, and are mately Young’s values (loc. cit.). It will be seen from the Table that the values of Nex, at. the boiling-point agree very well with the values of the latent heat of vaporization. At the boiling-point, there- fore, we have, in equation (6), H+hk = 0. For ‘“ normal” or unassociated liquids h=0, and hence 17 H= | (C,-c,)dT = 0. 0 a Substance. Nitrogen Oxygen ey ee a Methyl acetate Fluor-benzene...... Propyl formate .. Propyl acetate Ethy! propionate.. Mlethy! butyrate... MMOTIC ............ es EMEC) 2 owas eae cle d eee eeesec eas] EH wass.--......., | ul im SMTUTA, 5.50.20 Ethyl acetate ...... | hexane ............ Nae Methyl oeesug » ee Im-heptane ......... | Methy! isobutyrate| : Chlor-benzene...... Bromo-benzene ...| Todo-benzene ...... Mcetic acid... ...... BMMEAMEYS 2 cevclecccce =) Ethyl alcohol ...... Methy}! alcohol ... 1 Dewar. 3 Keesom and Onnes (1913). a. ats. (22°4 litres)” 00042? 002762 00272 00260 027982 0750 03688 03915 02355 02412 03302 03316 04383 05383 04272 04388 04330 04148 03945 (11). ‘06876 05659 05582 ‘05577 05339 05240 05926 08351 ‘05580 (05692 06592 03732 ‘0118 ‘02617 02512 01959 TABLE. a. Cok : e.c. per| cals. gram. |per gram. 1429: | 89°47 1:230+ | 3482 *857* | 37°75 "7124 | 27°95 1-4766 | 227-7 1:00 47°15 1-431 56°08 1°622 59°96 1-059 75°22 "822 61:8 17133 64°64 1:136' 61°88 1-20 57°39 1-633 54:12 | 1-053 53°61 1-199 57-46 1-189 57°17 ‘676 316 1:233 64:03 1633 51-26 1:257 52°67 1°258 519 1-240 52-55 1:239 50°3 1-283 58°75 5021 | 21°12 1621 48:2 983 47-54 ‘769 36°5 ‘633 30°5 1068 | 118-2 1044 | 424-7 1321 71°56 1:360 | 106°3 1324 | 1760 0,142 0,150 ‘0,166 I 0,.35 0,138 ‘0,156 0,155 0,159 | 0,156 0.155 09125 | “Og 141 0,154 -0,113 0,114 0,128 | -0;,766 0,149 0,125 0,121 2 Kaye and Laby’s Tables. 0,11162 ‘0569+ 04264 ‘03854 ‘0,176 ‘0,186 ‘0,169 ‘0,163 Onl ag ‘0,126 ‘0,159 ‘O, E58 0,152 02159 1:227 1:438 1-362 1°385 1-419 1-472 1-52 1-511 1-491 1-407 15138 1-528 1-521 1544 1-503 1-530 1-586 1-470 1-487 1-582 1-578 1593 1°58] 1-565 1-478 1-542 1606 1-506 1-490 1-516 1-484 1-286 1:492 1:44] 1-402 + Baly and Donnan, Jour. Chem. Soe, lxxxi. p. 907 (1902). 5 Eucken, Ber. deut. phys. Geselisch, xviii. p. 4 (1916). ° Lunge. : ® Marshal and Ramsay, Phil. Mag. (5) xli. p. 28 (1896). ® Brown, Jour. Chem. Soe. Ixxxiil. p. 987 (19 10 ‘Tyrer. 7 Estreicher. 03), Ixxxvii. p. 265 (1905). Phil. Mag. Ser. 6. Vol. 44. No. 261. Sept. 1922. r vap. observed. cals. per gr. 108°5- 113-23 49°83 53.8 37°55 3217 594 The late W. Gordon Brown on the At the end of the Table are given some results for liquids ordinarily regarded as associated. It will be noticed that in the case of water the value of H+h is approximately zero ; in other words, h+H must be nearly equal but opposite in slon. EN nt between the values of Aex, and Avap. 1s not found at temperatures other than the poiling-point. Thus tor ether|at O° C., Acx = $5'D cals..s -Avap, = 9 470 Calcap aan from equation (4) it is easy to show that the slope of the Nex. Curve is given by ht Nee a bal aT ae a ee Bi Now at the. boiling-point, «T is approximately *5 (wde Table above. Hence Ee a e pe — —d,VqnVqsd )—4( (2d,7 /(K)] aaa (E+ Vq,,B), oy dite. bel Pp eek vo aerial (14) where the summations include all values of the suffixes n, s, the differentiation of terms such as (—dpVQnVQndn) being performed by means of (11), since (—VqnVQm) is a self- conjugate ‘operator ; ; and that of cross-products, such as :— (—d,,VqnVq,d;) by means of (10), writing a= —Vq,Vq.ds. (| GC. A D Thus, if in the figure the unit segment is removed from the position AD (at which (14) has the value —(E+ Vq,,B) ) to the parallel position BC (at which (14) has the value —(1+6ry) (E+ Vq,,B), AB=6r), then the total increase in L is given by 6L=—éry.dd,,(E+ Vq,,B). It will now be convenient to suppose (as we may without loss of generality) that the mth set consists of but one tube, so that 6d,,=4,, and is in fact a unit vector. Then 6,L=—ory.d,(E+Vq,B), . . . (15) and in applying the axial differentiator 6ry we must re- member that neither d,, nor q,, as they occur explicitly are to be considered variable. But to preserve the continuity of the tube we require to introduce the segments AB, CD, as shown in the figure, so Faraday-Tube Theory of Electro-Magnetism. 599 that, again applying (13), we have the change of L due to this cause ole0, 7 Or Va_B),) 2). (16) in which q, is variable (but not d,). Hence §L=6,L+6,L=oer[ day. (E+ VgnB)—V .dn(E+ VanB) |, q, Varying in the first term only, and d,, not at all, and finally on = aay (E+ VqnB)—V .dn(E+Vq,B) . (17) with the same convention. In calculating the momentum term ob we have r=qp.- or Then by the method employed above in aa (14), since T is symmetrical in q,, and d,,, ox yy = mage ° ° ° ° . 18 Aah Vd,,B (18) | . . oL e e This will be the value of F when d,, is a unit length of tube, but in performing the complete differentiation to time in (7) we must remember that any length of tube will in general be continually varying in direction and magnitude. It is clear that d dt i oaegpemen | A - Ams a Nae eaae MO peeeie (19) since the rate of change of a segment of a straight line, as AD in the figure, will be the relative velocity of its ends (vectorially) ; “while, of course, if q, expresses the velocity of any point of the tubs, as A, the velocity at D will be (1+ADvy. )Qm, where AD is the vector element. Thus dol d aan oa = VdnV -Gm)B+ VdnBt+Vdn(dnv-B), - - (20) where B is the rate of chan ge of B at a fixed point coincident with the moving centre of the segment, qnV-B being of course the term in the rate of change due to motion of the segment with velocity qn. 600 UWP Tate Worden Brown on the Equation (12) is therefore by (17) and (20), Vn -Im)B+ VdnB—Vdn(dnV -B) —d,V - (E+ Vq,B) +7 -dn(E+ VqnB)=0, . (21) d,, and q», being constant in the last term, and 7 operating forwards only. i or In carrying out the simplifying transformations we may drop for the moment the suffix m. From the last two terms we have, in part, —dy.E+y.dE=+VdVvE = + Vd curl Ey eee) From the remainder we find VdB+ V (dy .q)B+ Va(qy.B)—dv. VgB+ v,.dVqQB, = VdB+ V(dy .q)B+V .d(qv.B) (—V(ay .q)B—V .a(dy.B)+y1.dVqB, =VdB+V.d(qvy.B)—V.q(dv.B) —V.d(qv.B)+V.q(dvy.B)+ Vdq. vB =VdB+ VdadivB,. 0 4 where the suffix restricts the action of vy to the vector carrying the same sufhx. Equation (21) then reduces to Vd,;(curl B+ B+ div Bi). =) Dates) Now d,, will have different values according to the different directions of the various sets of tubes; hence (unless ail.the tubes are parallel) we may write ‘curl }B+q,,divB=0.. 1 Yo 907 Qe) From this, since q,, is the velocity of any set of tubes, unless all the sets have a common velocity, we must have div B= 0} hee a 2s Sea er and thus “enrl R= Bie oe eer ee (27) We have now shown that the first four laws of the ordinary theory of electromagnetism are consequences of the assump- tions which have been made. It may be observed that whereas, in the proof of the first two laws (3) and (4), no departure of importance is made from the method of ‘ Recent Researches,’ the proof just given of the laws (26) and (27) is quite different from that adopted in that work. This is Faraday-Tube Theory of Electro-Magnetism. 601 rendered necessary by the purpose of the present paper, which is not to deduce the properties of the tubes from the known laws of electromagnetism, but to show that, given the tubes with the (essential) properties assigned to them by Sir J. J. Thomson, the laws of electromagnetism follow. 5. It remains to discuss the forces acting on the electric particles. Referring to the figure on p. 598, let B be a particle at the end of the tube B,C, D. ‘Then the change in L due to the displacement of the end of the tube from B to A (introducing a new segment BA), is by (13) Boe Ee VOB ere sy fe. ues C28) since éd,,=AB= — or, B being the positive end of the tube, and thus equivalent to a positive unit of electricity. Hence the foree acting per unit charge moving with velocity q is © | ie OTE Mate 3, Aa era a Ek 45) the Fifth Law of Electromagnetism. 6. The definite dynamical assumptions of this theory enable us to examine very thoroughly such questions as the stress in the field and the mechanism of radiation. Heaviside * has given a general discussion of the problem of stresses from which it is not difficult to deduce the following general result :— Let yr be the operator of Maxwell’s stress, yy=E.D+H.B—3(ED+HB), . . . (30) where any vector operand forms with D and Bscalar products in the first and second terms. When this operand is a unit vector N, WN is the stress on the plane perpendicular to N. Let be the stress derived from yy, by putting for E, E+ VqgB, and for H, H— VqD, namely w= + VqB.D—VqD.B—43(VqB)D+4(VqD)B =yW+ VqB.D+ VDq.B—DVqB =Wo+VDB.q. Z : ; ee 2 é . (31) by mere vector transformation. Then if Nis unit normal toa surface moving with a velocity q at any point, YN is the flux of momentum through the surface in the direction opposite to the positive direction of N, per unit surface per unit time. * ‘Electrical Papers,’ vol. ii. pp. 521 et seg.; also Phil. Trans, A. 1892 |. | | i: | 602 The late W. Gordon Brown on the To see that this is true we have only to apply the theorem of divergence; in the first place we note that since ol. — Vd,,B, c 0 iY ot ° : (18) summing for all values of m we have VDB equal to the momentum per unit volume. Bat yv= 2 VoB, Me a result easily deduced (Heaviside, loc. cit.) from the circuital laws, and usually expressed in words by stating that Maxwell’s stress gives rise to a translational force per unit volume equal to the rate of change at a fixed point of the momentum per unit volume (the absence of electrification being assumed). We are thus entitled to say that YN is the flux of momentum per unit area of a fixed surface. Now it is clear that VDB.QqN is the flux per unit area due to the motion of the surface with velocity gq. Hence w is the general operator giving the flux of momentum. The equation of rate of change of momentum per unit volume at a point whose velocity is q is 7 yo= 2 VDB+ vq. VDB a 2 VDB+ay _DB+VDB.divg, . . (83) the first wo terms giving the rate of change of density of momentum at the moving point, and the last term the rate of change due to expansion at the rate div q. This flux of momentum > is partly due to convection, and partly to be ascribed to a stress. It is interesting to note that if all the tubes were of one set, we could determine the stress by simply putting q equal to this velocity. We should then have H= VqD, and the stress would be =(E+ VqB).D—3(E+VqB)D =F.D—LFD =F. D+L(B=—ED). a 2) 3 In general the stress operator will be obtained by sub- tracting from yo the operator —3(Vd,,B.qn) which gives the convective flux of momentum relative to a fixed point ; thus a Faraday-Tube Theory of Electro-Magnetism. 603 the stress is p= Wot+d(Vd,B. oer (35)' =E.D+H. B—1ED—1HB+SV a ee eis Va. ai: B +3(Vq,,dm)B =E.D+H.B—iED—+HB+Vq,,B.d,—H.B+HE = {E+ Vq,B.d,} —4ED+iHB. FEM Leen eres (DD) From (35)' we see that the stress coincides with Maxwell’s stress when there is no convection of momentum relative to the (so-called) fixed reference frame ; and from (35) that it consists in general. of a quasi-tension equal to E+ Vq,,B per tube of the mth set together with a hydrostatic pressure 4(ED—HB). The torque per unit volume is seen to be i) a gp’ =s=— +V(E + VqmB) cee = + > VdinVqmB =-+ >VanVd,,B ai >V (Vqmdm)B SSeS MeN Bt a ad ats ts Moe (3B) the last expression being the rate of change of moment of momentum about a fixed point due to component of velocity perpendicular to the momentum, familiar in the hydro- dynamics of the motion of bodies in a fluid. 7. The flux of energy also consists of two parts: the convective flux due to the motion of the tubes, and the flux due to the activity of the stress. To find the convective flux we require to localize the energy in a manner rather difficult to justify. The whole energy per unit volume may be written INB+1ED =+4%3d,,(E— Vq,B). Poe tke Th 3 (37) Then we may suppose the part d,,(E—Vq,,B) of the energy to be moving with velocity q,, and so on. The total con- vection of energy will therefore be $3d,.(E— —VqnB). Qm- e. berRiise) Wurm (38) To find the stress- pies flux from (35), consider first the term (E+ Vq,,B) .d,,; the appropriate velocity is clearly Qm, and the flux (by Heaviside’s method) —Qn(E+ VgmB) .dn= —QnE. dn. Again, we may write the second term — LED + SHB=—34(54,,)E—(SVandn)B = —42d,,(E+ VanB), and it seems permissible to write the activity flux due to the term —34d,,(E-+ Vq,,B) as +4q,.d,,(E+Vq,,B). Hence the 604 "The late W. Gordon Brown on the total activity flux will be a : —S4qnE.d,—$dn(E+ VanB)}, . . . (39) and the whole flux, adding (38) and (89), ne. W =324,,(E— VanB) «Gm —2OmE «dn +320,,(E + VanB)n dee Qn —AmE . din) . = VES Vq,,d, == aR RT erie Le te al 8. Since we have By ‘that this theory leads to the ordinary equations of the electromagnetic field, it is un- necessary to give a separate proof of the uniform propagation of disturbances with velocity 1/ mK. It is perhaps as well, however, to examine shortly the mechanism of propagation, particularly since the mental picture of electromagnetic radiation afforded by the theory is in many respects very satisfactory. N. Campbell gives a short discussion of the question, and shows that a tube at rest may be compared to a flexible cord of linear density »D under a tension D/K ; the square of the velocity of propagation of transverse disturbances being then 1/uK by the elementary dynamics of cords. To extend this result to the case of a tube having a general velocity v perpendicular to its own direction, we have only to remember that, by equation (39) above, the stress to which the restoring force i is due will now be the. quasi-tension E+ VqB, where q is the velocity of the tubes, of which we ‘shall suppose that only one set need be taken into account ; and with this last assumption we may drop the suffix m and so write . d B =pVqD, k= K : The d component of E+ VqD is the only effective part of the stress, and its magnitude is given by d (E+ VqB)d, = (x + pVqVad ) ie where d, is the unit vector parallel to d, or d=dd,. This equals 2 (1+ wKd,VqVqd) d = x (l-wK(Va.a)"} vw? ! lo ee aie, il) Ale eukK=1. Faraday-Tube Theory of Electro-Magnetism. 605 The linear density will remain pd, so that the velocity of propagation along the tube will be 4/c?—v?. Since the tube itself is in motion with velocity v in a perpendicular direction, the propagation of the disturbance in space will be with . . . . . . UV . velocity ¢ in a direction making an angle sin~"- with the tube. When v=c the disturbance will not be propagated at all along the tube, which will lie in the wave-front ; and the traction (E+VqVqD) will vanish. 9. To take into account a general velocity of the tube in the direction of its length, let us restrict ourselves to plane- polarized radiation. We shall take the w-axis in the direction of propagation, and the y-axis in that of the disturbance. Since we are dealing only with transverse vibrations, the velocity of the tubes in the direction of the ray will be constant from point to point along a tube. Let wu be this x-component of velocity. Also let (#,y) be the coordinates of a point on some particular tube at time ¢t, so that y is a function of z andt. Then the whole y-component of velocity of the point will be se eee OF ? oS pea Nin Ey eee See (42) It is obvious that the shearing motion perpendicular to the a-axis of the tubes in their vibration will not affect the number of tubes per unit area passing through a plane normal to the x-axis. Thus the quantity d,, the x“-component of electric displacement, will be constant at a point on the tube, or fe Or oa (2 +u <.\d-=0. Also, if d, is the y-component, we shall have dy _ OY de OF And thus ips oy, P= dy + dy? DE | Oy ie ; pe ia las) ; ; Sere 2 os (43) The momentum per unit length along the tube is Vd,B=,Vd,Vqd 4 | . d—d ° qd;) . 606 The late W. Gordon Brown on the Multiply this by $ to find the value appropriate to unit length along the #-axis, and, taking the y-component, we have war Ge F530) 45a ate (SE 44g) ef pd, mua.| (2 +e a ae (a y ut) OU | Ot Ox Ov ae Ox [Se ++(S2) -(82) } +32 +(82) 2-2) f Side ol Hence the rate of change of momentum in the y-direction per unit length along the #-axis is O20) By Oy O°” D (5: tego )adedy eee Set aapef CS) The force to be equated to this arises from the quasi- tension B+VoB= — +yVaVaD d 2 = ted. ad—pd.a°, of which the y-component is cd SY 4 w(t 4) Sud, + (SY +See) Kei ow fo Ot. 0 foe —ud.2" 4 2+(8 a (22 4a out fe OF oy, OY Oy mie 3p HS; Tees ud os 2 il = hel + pd OU wan ie beam ae MMT fics ma » (46) Differentiating with respect to # we have the force per unit length a,| oh tweed ay a Faraday- Tube Theory of [lectro-Magnetism. 607 Equating therefore expressions (45) and (47) and dividing by dz, we have 10% By | Oy | OY K 32 THY 30 ot aco | Ot” or O’y 2oy e Ae ok wating joy C2) exhibiting the uniform propagation with velocity ¢ inde- pendent of the general motion of the tube. The relation between the electromagnetic disturbance and the displacement y of the tube is easily seen to be given by eyo OY By= 3 = Kd, Se enc Ov ee at = . . (60) But while in plane-polarized radiation the displacement of the tube from its normal position is thus perpendicular to the plane of polarization, in circularly polarized rays it is easy to see that the reverse is the case. 10. The intention in presenting the theory of Faraday tubes in*the present form was to suggest possibilites of modification which might explain various phenomena of which no entirely satisfactory electrical explanation has been given so far. In making attempts of this kind we may, for instance, take advantage in various ways of the fact that the electric dis- placement has been considered as a mean value taken over a small, but not infinitesimal, area. From this point of view the Maxwellian theory is microscopical, and a more micro- scopical theory may be what is required in various regions of modern physics. Again, the present theory rests on the fecalieaiion) of electric and magnetic energy as functions of D and H and on the derivation from these of equations of motion. Hence it would be comparatively simple to estimate the effects either of a modified distribution of energy, or of substituting any different hypothesis for the principle of action. Lastly, quite a variety of hypotheses are possible as to the exact nature of the electric particles. 11. It will be observed that in describing the properties of 608 The late W. Gordon Brown on the the tubes of force we have so far assumed that two oppositely directed tubes at the same point exactly cancel each other in their effects, if they are moving with the same velocity. Now, just as the electrical theory of matter explains all the phenomena of neutral bodies as due to the existence of the equal mixture of positive and negative electricity, which on the two-fluid theory was supposed to have no recognizable physical properties, so on the lines of force theory we may perhaps speculate with advantage on the possibility of ex- plaining by means of properties of equal mixtures of oppositely directed tubes the phenomenon of gravitation, which seems for many reasons to be on a different level from the ordinary electrical phenomena. Let us consider the potential energy of such a mixture of tubes. So long as we choose an element of area large enough to include many tubes, the density of energy 4ED must always vanish ; but as we take smaller and smaller elements of area, there will be an increasing probability of the number of tubes passing through it in one direction being not quite equal to the number passing through it in the opposite direction : in other words, what to ordinary microscupic electrical measurements is a mni- form absence of electric displacement may consist of alternate regions of opposite displacement so smali that only the mean field of a considerable number of regionsis measured. Such a field would have positive potential energy ; but since the more closely the tubes are packed, the smaller is the element of area we can take without considering this effect, it seems reasonable to suppose that the effect will become smaller the more numerous are the tubes of either sign. Not improbably a mathematical form might be given to this hypothesis which would explain and locate the energy of gravitation. Let de,, —deé,; deg, —de,, be pairs of opposite charges ; 11, 72 the (small) distances apart of the components of each pair ; and R the distance between the pairs. Then if the hypothesis could be so formulated that the potential energy of the system would include a term of the form —yde,*de,* 3 Tro where ¥ is a positive constant, the law of gravitation would be completely satisfied, and gravitational mass would be identified exactly with electromagnetic mass ; for Vy is proportional to the element of electromagnetic mass due to two elements of de,, — de. Faraday-Tube Theory of Electro-Magnetism. 609 This last question is of some interest in the theory of atomic structure ; a number of writers have laid stress on the importance of mutual electromagnetic mass, and in particular Harkins and E. D. Wilson * have used this phenomenon to explain the departure of atomic wezghts from whole numbers. It appears, however, that such an explanation could alone be valid if mutual mass were ponderable. 12. The theory of Faraday tubes might possibly be em- ployed with advantage in other investigations connected with atom theory. Sir J. J. Thomson + has made several sug- gestions of this nature; his conception of the electron as possibly simply the end of a single Faraday tube would, of course, have very important consequences if adhered to in any theory of atomic structure. Again, if we suppose that electrons and positive nuclei have the property of excluding the tubes of other electrons and nuclei, the attractions between particles of opposite sign would become a repulsion at very small distances. Or we may suppose that some or all of the tubes of an eiectron in an atom simply end at a nucleus, instead of spreading equally outwards in all directions ; and different states of an atom, with different periods of vibration, might arise according to the number of tubes so connected. ‘Suggestions have also been made as to the application of the theory in connexion with a possible discrete structure in radiation f. CONCLUSION. 13. It has been shown that the general equation of the Maxwell-Lorentz-Heaviside theory of electromagnetism can be derived as macroscopic consequences of a simple dynamical theory of Faraday tubes. This theory also gives explicit and non-contradictory expression to the ideas of electromagnetic stress, momentum, and flux of energy, and an electromechanical picture of radiation explaining the law of uniform propagation in spite of the motion of the source. A number of suggestions are made as to applications to the theory of gravitation and other problems. Hawke Battalion, Royal Naval Division. * Phil. Mag. Nov. 1915, p. 72 Tt Phil. Mag. (6) xxvi. p. 792, t Jeans, “Report on Quantum Theories,’ Proc. Lond. Phys. Soc., 1915. Phil. Mag. Ser. 6. Vol. 44. No. 261. Sept. 1922. 2R ) 610 J LV. Molecular Thermodynamics. III. By Bernarp A. M. Cavanacu, B.A., Balliol College, Oxford *. | SOLVATION OF SOLUTES. . ee partial solvation of a solute—that is, the. com- bination of some fraction of it with the substance which, in the free state, constitutes the solvent—to form a ‘‘solvate” or “solvates”? 1s a phenomenon probably of the very widest prevalence in solutions, particularly in our more common and valtable solvents. It has unfortunately been too readily ignored, because in dilute solutions its effects are not of the first order of magnitude. They are nevertheless considerable, in many cases much more con- siderable than has generally been allowed for ; and, in any attempt to pass beyond the region of the most dilute solutions (say, 55), the question of solvation of the solutes must receive very serious consideration. 5 In the present treatment, it is to be understood, the “solvates ’ considered are true chemical compounds, at any rate in the one sense required by molecular thermo- dynamics f, and the conclusions reached are not to be expected to apply in any degree to any other “ associations ” or “complexes” to which the name “solvate” might be given. . The following are the general circumstances considered here :—— Any number of “solutes” dissolved in a complex ft solvent are indefinitely solvated, without dissociation or association—that is to say, each forms one unsolvated molecular species and any number of solvate-species by combination of one unsolvated molecule with different amounts of solvent. The residual solvent not so “absorbed” by combination with the solutes may be called the “ free solvent ”’ §. The separable or experimental solutes have been called the ‘“unsolvated” solutes, and it would appear a little paradoxical to say that these may be solvates ; but, in fact, * Communicated by Dr. J. W. Nicholson, F.R.S. + See second paper, Phii. Mag. xliv. p. 229 (1922), 1st section. + See second paper, Joc. cit. The case where the solvent is simple is included as a limiting case. § In the previous paper the term “solvent” was used to signify ‘‘ free solvent,” since the combined or ‘‘ bound” solvent was not there under consideration. ¢ : we a" On Molecular Thermodynamics. 611 the treatment below will apply in full and without alteration - in such a cuse, if a lower solvate be regardedas a “ negative” solvate of a higher. The following alternative and more general statement of the problem will make this clearer :— Yhe molecular species present in the phase or solution are the several solvent-molecular species which ‘together constitute the ‘free solvent”? and a number of solute- molecular species. Now the latter can be grouped in such a way that members of a group differ in composition only in the amount of solvent per molecule, and can be regarded as positive or negative solvates of one another. One member of each group appears in the present discussion as a separable or ‘‘ experimental ”’ solute-—a component under the conditions of experiment ; and the (positive or negative) quantities of solvent by which the other members differ from this one are, and are alone, regarded as ‘“‘bound”’ solvent, which together with the “ free ” comprises the “ total solvent.” In the “molecular” expression for yw, from which we started in the previous paper, every molecular species appears as a component. In the “ experimental ” expression obtained in that paper the free solvent appears as one com- ponent only, and in the present paper, starting from this result, which we may refer to as our “ original “ expression for wp, we shall obtain an expression in which the “ total solvent ” and the ‘‘ unsolvated solutes”’ are the components. It is clear that problems, such as dissociation and com- bination of the solutes, excluded from present consideration, can be treated as further stages in this gradual reduction of the components. Let M, be the mass of free solvent considered, and M,’' that of the total solvent, the latter but not the former being an experimental (known and controllable) quantity. Concentrations referred to the latter may be called ““ experimental” concentrations, DY, Peas) feet te. Re es ae =p? C MiGr ane (1) while those referred to the former, mee eS Site N cs = M,’ Sie M, = M,’ . = oy Wie (2) may be ealled ‘‘ true”’ concentrations. The latter alone were used in the consideration of “ complex solvents.” It will be a convenience making for clarity, to obtain 2K 2 612 Mr. Bernard Cavanagh on our results in the first place in terms of the “true con- centrations,’ introducing the “experimental” concentrations afterwards. Of the total number n, of gm.-molecules of the solute, (s), put into the solution, let the fractions ws,, as,,...... , go to form solvates containing respectively M,,, M,.,...... > grams of solvent per gram-molecule, and let us write X, for 22;,,, and a; for 2s, Ms, ; then clearly (from previous paper) our ‘ original ” expression for sf is ee Mo[gxtR| = ae (140) | c= vim f a -X,)[.—R log gaa m — ~ Css vod + Das, [@.—R log aera } +M,Gs 2) MoC and is of the form fo) fo) fo) ap = My Su, tet Xe) se ai BEng Ss, s (4) the independent variables being My, n.(1—X,), nets, Ns& 504 eoceces The ‘“‘experimental”’ expression for yw, now required, must have the form De, vr = M,’ oM,! +3n aay ae? (5) the independent variables being My’, n,, ...... In changing the variables we make use of the equations given by the chemical equilibrium governing the various stages of solvation, viz. : OV Pot Ob Ons, < ONsy jas ill; ne Ons. Ones 29 OM, etc., whence, of course, OY uy OM ov Se =a Sis “SOM, =) Bi ar tade KONE AC (7) Molecular Thermodynamics. 613 and similarly ON teres Es, tnt, A1—X) whereby the (somewhat lengthy) full differentiation of (3) with respect to the new variables reduces to ow gege, er (8) Ov _ ov (9) OM,’ OM,’ Ov _y, OE Ch oh a Sn, 7g, 1 KS tes -. (10) = 2, (11) equations of obvious physical significance. Now (11) with (3) gives B3 +R log | = ¢,,—R log (1— Xs) +R log (= Hf mig()) BGH G2) and Lt [S¥+ Rog cs | = ¢y—B log (1—X,,) = $s (say), (13) since X,,, or ie X,;, depends only on temperature and pressure. Then B¥ + Riga] = dt | a(R tog], 1. é. On the one hand, of course, Sie ela in. as MgO m i Hx] +Ge! =, (15) but, on the other hand, from (8) and (10) we get a(3t) = 28 (Se) +a—Ko4 (SF) —aa( Sa, 9 ov be— —R loge Cs + +f a3 oF R log cs]. (14) ee el 614 Mr. Bernard Cavanagh on and, referring to (3), while remembering La, +(1—X,) = 1, Ya,,d log #,, +(1—X,)d log (1— Xs) = 0, we obtain d [SY + R log «, | mH [a ee (S+mc)—Sa aie +m) | + Xw,,dG,,’ + (1— Xs) dG,,/ —a#sdGm'; (17) whence (14) gives, as alternative to (15), Ons Mow = + Te) C=0 (18) where Gr=\ 10x dG./ 42nd,’ = .dGa Jon C=0 Comparing (15) and (18), we obtain — Pulivees Ea: te R ee Ke = R| : —d log (1+ mC)—(G.—G,, ), 20) or 4 Rd log (L—X,) = R Sd log (1+mC) = + [a,dGmw’ +X,dG,,'—d#,,dG,,'], (21) which could be obtained independently from (7) and (8) with (3), and then used to get (18) from (15) ; but the above derivation from the physically significant equation (10) appeared more interesting, and brings out an analogy with the previous problem of “ complex solvents.” When there is present only one solute (s) (but any number of solvates thereof), (18) can also be obtained easily, by means of the Gibbs fundamental relation, from oF . 0 The general terms being G, in (18) and G,,’ in (i5) we see that the division into “linear” and ‘* general 7” Molecular Thermodynamics. 615 terms is different in the two expressions for a the s difference being the quantity Co 1 { Bee ae eM Cac). « (22) C=0 The form (18) will probably be the more generally useful, possessing the advantage that the “ linear ” and ‘‘ general”’ terms separately are connected by the Gibbs fundamental relation with those (respectively) in the simple form ov f 5M,” a= 2 st = [otk | as =¢ sh eee ae ote Gy we obtain a function (say, ai of the “ experimental ” concentrations (O’,cs’) and of the various a’s, as inde- pendent variables, and clearly Se eee = 6, PME) 430, e) +05 BH), Ons = (l—X)G,’ +32.,G,,/—a:Gu',... >. 1) 2 aes for which we may therefore write G,’', so that M,'G" = M,'Gu' + dn.G.". 2 2 ees) On the other hand, a ba me Se le ae = M/ naz Gey oe (see aa | eG es [ Gs, — Gs, —Mi,Gau'] = Gz,” (say). = (84) (19), (32), and (34) then give Gs = G,!’- Gao rs C=0 mG: p) oGe | =Sa Sle aie me SS. C=0 and therefore from (28) G= ode 2 Ga, xs, ele = Ows, dz, iGo ox (35) and (36) being plainly in accord, as they must be, with (29). Molecular Thermodynamics. 617 Our final “ experimental” expression for yy, in terms of “ true’’ concentrations, is ce 1 pS wr = M, [out Rf —d los (1+: nC) | C=0 Cs + ins [ oR flog =——— mM = + (Ba log (1+ 0) } +e, (37) C=0 where G has some form Gree (ales earn one cat tae) The “general” terms in the original equation (3) were M,G'—that is, M,'G"’; so that the quantity ; reins OG! v M Vins ati SSE Sn. | 3G., dfs, or Ene 71> ae Te GD) C=0 has passed from “‘ general ” to “ linear ” terms in w& ; or, in b) cy, the quantity $s Gs'—Gs= | Ge, "dt, . + + - (40) C=0 We may recall at this point equation (22), which is now Gs—Gs, = | 2asdGe,", - - 2 + (41) C=0 : and note that, disregarding (more or less fortuitous) can- cellation, the condition for the vanishing of (40) and (41) is qualitatively the same, viz., that G’” shall be independent of the extent of solvation of the solute (s)—an obvious particular case being ‘Perfect Solution,” where G" vanishes. The lack of practical significance in such a qualitative statement is illustrated and emphasized when we observe that quantitatively the condition is very different in the two cases. The two differences (40) and (41) are, in fact, complementary parts of the quantity eee eee fe! (aay 618 Mr. Bernard Cavanagh on and since sith { Co Gan te dix ce whereas ( dz,, will be only some fraction—generally, as wic=0 we shall see, not large—of «;,, a simple graphical considera- tion will suffice to show that (41) will generally be much greater than (40). The practical significance and value of the quantitative statement (39) is further illustrated in Appendix II. The analogy between the modification of the “ general” terms, expressed in (39), and that found necessary in the previous treatment of ‘‘ complex solvents” is plain. In the latter we kept fA ; Gres G; ’ whereas in the present case we have kept Gu mae Gy’, but, of course, since we started in the present case from the results of the previous paper (and G’ in this paper corre- sponds to G in the previous paper), the two modifications of the ‘‘ general” terms in the ‘‘ molecular” expression are, as it were, superimposed. More detailed treatment than the above of the “ general ” terms will be possible only when something is known about their form (see, for instance, Appendix IT.). For the evaluation of the linear terms we observe that. _ a; must be expressed as a function of the concentrations of the solutes. An Expansion of as. Consider, first, a solute which forms only one solvate. In this case we can write as, M,, etc., instead of a,,, M,,, ete., and, moreover, we have ag = Ma, X; = Ws : ° 0 ' 5 (43 ) ee and equation (21) assumes the form, Rd log (1—a,) =as Es log (1+mC) —dG,, ale (44) ee Molecular Thermodynamics. 619 or if we write == ; 4 ; aay 1 | B\ y= [uf = log (1+ mC) — Bors Sat (AOD then d — am Sa fy —— wo). e . . ° 4 If now we write w, as a series of ascending integral powers of y, we can show that a few terms only need be taken for our practical purposes *, so long as y is little greater than unity—that is, we obtain a practical expression for ws which will serve up to very high concentrations. Putting then Us= As, t+ by t+ boy? + bsy? + 2... +Ony”... by straightforward substitution in (45) (and equating co- efficients) we obtain 0, b,, ...... in terms of a@s,. A simple relation between these coefficients enables them to be written down very easily, viz. : 1b, = by ao the first coefficient ), being zs,(as5,—1), and being a factor of all the others. The alternate coefficients are also divisible by (a’s,—34). : The numerical values of the coefficients depend, of course, on that of ws, which may be any proper fraction (positive). Their maximal (absolute) values can be obtained and shown to decrease rather rapidly. The higher members of the series have, of course, several maxima, hut the greatest maximal (absolute) values diminish by alternate long and short steps, the. following round values being sufficient for present consideration : b,. be. bs. ve De. De b.. 1 i i i 1 uh er + 22 48 300 480 4400 - Taking } per cent. as our standard of experimental accuracy, we see that the series as far as 7°, Vs Xo + by + boy? + bsy?, : . : ° ° (4 7) will suffice until y closely approaches unity, when the 7 term will be just appreciable. * That is, the remainder after a few terms is negligible experimentally up to high concentrations. We may conveniently call this “‘ converging practically up to high concentrations.” 620 Mr. Bernard Cavanagh on We have then a as a practical function of y, which is a function of the concentrations of the solutes (see (44)), and if we know the form of Gz, we can readily complete the as evaluation of ie dlog (1+mC). Mh C=0 Us R First, let us assume that =; G,," can be neglected and write y=M| J dog Gmc): C=0 3 . From the previous paper we can then write, y= M,C[1—$a,C + 4a,C? —443C*], and from (43) and (47) we then get t= a+ B,C y,C? +6,05%.. . 5.) eecean Where as Mien (3S (b.M,? mae $y b, M,”) 9 49 Be= M275), onan Osan ie eee eS which may be expected to hold up to a value of C rather higher than ir (since y is considerably less then M,C at high concentrations), that is, in aqueous solution for example, up to a total concentration (“ true’’) of at least twelve-molar (12M.) when the solvate is a pentahydrate, or six-molar (6M.) when it is a decahydrate. When ae! is not negligible the above will cease to hold exactly, but it may be possible to express =0Q5 (with a R negligible residual error) as a function of C, the total solute concentration, and if this function can approximately take the form of a short series of integral powers we shall get a result in the above form, (48), but with departures in the values of the coefticients y, and 6,. When the solute considered forms several solvates, the problem of obtaining «, as a function of the concentrations is more complex. Hquation (21) takes the form d log (1—X,) = 2a, [= dlog (1 + mC) — but we now have to use instead the series of equations (6) dGe,,” 3 (90) Molecular Thermodynamics. 621 from which, as already remarked, (21) might have been directly obtained, and which can be written = Mint h — Lay 2 4a,C? —ta;,0*)dC— cen 1X, Hit (BU vs d log =M,1—},0 + Ja,(?#—3a,0%)dO— 5 dGe,, (and so on). Again, we pass over the most general ease, where it is necessary to consider Ge"; etc., as depending in specific ways upon the particular solutes present, and we suppose a . . ° al (except for negligible residues) G,, ", etc., can be accounted for as functions of C. If these functions can be written approximately as short series of integral powers, then the above equations will take the general forms, Fee. NB Y = [got mC + 920? + g3C® Poses |dC | d log En fee Re eee ante , (52 dog —* = fg! + 9/C-+g/0r+g/0......Ja0 (7 ©” ZS | ete. J WHERCE 25, 25,5: .2-5-: (and so e,), can all be obtained in the form of ascending series of integral powers of C, by a procedure entirely analogous to that used in the simpler case of one solvate, viz., by assuming such forms for a@,,, , and then determining the coefficients in terms of Gos Gy. Pas7- ee , by substituting in the above equations used of course simultaneously. | The interesting simple case where there are two solvates ae Gog. ean, be neglected, has been investigated and it has been found, as might be anticipated, that in the expression for a, a=a,+8,C+70C?+6,0?, . . . (53) finally obtained, the coefficients are quite of the same order of magnitude, as regards maximal values, as in the case of one solvate. They are conveniently obtained in terms of a a 622 Mr. Bernard Cavanagh on -series of symmetrical expressions of the form (My @e4- Ms), CMs, 7 2s, + Mig eM, 8a. tM eee where %, and w,, stand for the limiting values Lt ,, and Lt a,.,. C+0 C>0 The same procedure exactly, however, can be applied to the more general case (52), but becomes, of course, more complex and laborious as the number of solvates is increased. There can be little doubt that the maximal values of the coefficients remain of the same order of magnitude so long as Guo,", ete., are minor quantities as they will be in general ; so it will be assumed in what follows that «, can be expressed in the form (53), and (49) will be used in roughly estimating the ranges of validity of the results we shall ebtain. THE INTEGRATED LINEAR TBRMS. We can now evaluate the quantity {¥atog (L+mC) or f a.(1 a0! + an? —agC*)dC, c=0 : c=0 and, subtracting log (= + To (as evaluated in previous paper), obtain — log = at mig) + {2c log (1+mC) c=0 , =¢,C + $2,074 42,0? + 12,0) = (oa) ‘ where é=:(4s,—41), €3 = (Ys — 41 Os + Ag as, — a3), } (55) 09 = (Bs— ay As, + Gy), €y= (05 — A Ys + 428s — 434s) ( and (37) now appears in the more practical form ; - Lon | OM oY = $;—R [log ¢, + e,C + $e0? + de303 + ¢,C4] + Gs =u + RO (1—$a,0 + Jao? —30,0%) + Gu . (56) INTRODUCTION oF “ EXPERIMENTAL ” CONCENTRATIONS. We have now to introduce the “ experimental” concentra- tions (¢c,', C’) in place of the “ true ” concentrations (c,, C) in the expressions we have obtained. Molecular Thermodynamics. 623 Since MS= Mo a DNges, let us write (as we have already done) N for }n,, and also N2=Dd nas = >n; [a,,+BsC +ysC? + §,C? ’ and T= + BC+ 7C? 4 80%, so that oh zn = , B= et, Sly cy day (OO) Besa © a M,’ ML: and G,! = Cc = M, =i | se) =142,0+8C?+yO? +804. (57) If %; 8, 7. 6 are not to be determined, but are already accurately known, equations (56) can be used as they are, for the ‘ experimental ” concentrations can be translated into “true”? concentrations by means of a curve plotted from (57). We are dealing now with the more common case where Zs By ete., are not known. From (57), in a fairly obvious manner, we obtain 3 =1+7,C' + 7,0? + r3C'? 4-740", ° : (58) where pee r= (293+ 3%8+7), | 59) ro= (Hy? +B), y= (Got + 6%°B+ 4ayy + 26?+5 and ie we =1,C' +40? +140%+41,0"%, . . (60) where 1, =a, l;=(1a°+22,84+7), (61) =(2%7+8), 1y=(24'+ 3%)78 + 3%07 4+ 38°45) It is now a straightforward matter to eliminate C and c, trom the “ linear” terms in (56), obtaining Ov OM, ou + RO 1+4,0' + #0? + t3C'* + 4,04] + Gu, i 0 3 ( 9 (62) =~ =,—R [log e,' +#,'C' + t,'C/? + t,'0'8 + t,'0"4] +G, 624 | Mr. Bernard Cavanagh on where t=(71—$4)), (= [ 73 ay (1, + $13") + aor — tas |, to = (7, ee Oy: + 4 a9), ‘i= [%4—- ay (73 + 117) + Ao + 1”) — As? and | ty)’ =(4+h), ts = | e179 + €o7 + 403 + I], i (64) to’ =(eyri tdeotl,), t4 =[ errs t+ eo(72 +4717) +e: + te, +1.) ) The Gibbs fundamental relation can readily be applied at this point as a check on the detail. 3) A SPECIAL PROCEDURE, SOMETIMES NECESSARY. A difficulty arises, however, in the practical application of this result, as it stands, at high concentrations. We have seen that the series obtained for a, may be expected, up to sufficiently high concentrations, to “‘ converge practically” in the few terms given, and the same clearly applies to the e-series of (54), (55), and (56). In passing to “ experimental ”’ concentrations, however, we I / 0 have introduced the 7v- and I-series for M and log—— My M, respectively, which, when 2, and perhaps also 8 assume large values, will frequently fail at high concentrations to “converge practically ” in so few terms. A glance at the values of the r- and /-coefficients will show this, and also that the difficulty would vanish if % and @ were small, since y and 6 occur by themselves only in the first power. Now since (62) can be used at lower concentrations (with the ¢ and ¢’ series cut down to two terms, see below) to determine «) and 6 experimentally, we are at liberty to assume on turning our attention to the higher concentrations that a close estimate of a and at least a rough estimate of ® can be made, though neither, of course, will be accurate enough for these higher concentrations. Our difficulty can then be overcome by the following procedure. Let %', 2’, 7’, 6! be estimated values of Zp, 8,7, 5. (The procedure is stated in a general form—if y and 5 are not estimated 7 and 8! will be zero.) hen there will be an estimated value (M)"’) of Mo, and ‘‘estimated”’ concentrations (c'’, C'’) referred to it, such that o/ORey ML)! sl ce C’ Es My” =[1+4%)'0" + BO" +y/C'F450"]. . . (65) Molecular Thermodynamics. 625 Comparing this with (57) and the remarks which immediately follow (57), we see that we can easily translate ‘“ experi- mental” into “estimated”? concentrations and express our experimental data in terms of the latter, so that it is now only necessary to express (56) in terms of these “ estimated ” concentrations. Now &@ being really expressible as some series a=8, + BC+7O? +508, . . . . (66) of which we do not know the coefficients accurately, let this be equivalent to some series %=a,+ BO" +y"C'72+5"C"8, 2 2. (67) Then my a M,= M,'—N(@+80"+7'"C'?4+8"C'?). . (68) But from (65) M,’= M,'—N (a Peay 02 £9), KG) so that My= My’ —N [(Aa) + {AB)C" + (Ay') C0"? + (A6)C"7], (70) where Lag as (Aa) = (%—%'), (Ay!) = =F), ete.» (71) And then we have © en ° M sage i! =o = 9y,7 =1—[(Aa)0" + (a8) 0" + (Ay’)O"8+(A8\C"], (72) which can readily be shown to be equivalent to br le 2 , a =or= TL, =1+(Ae)C+As)C+ (Ay) C% + (A6)C%, (73) where (Ay) =[(Ay) + (Aa)(A8)] 5 (Ad') =[ (Ac) + 2(A2)(Ay) + (AB)? + (A«)?(A8)]. (74) On comparing this with (57), it is at once obvious that the result of introducing (¢,"’, C’’) the “ estimated ” concen- trations into (56) can be written down at once from equations (58) to (64) by simply writing ¢,’’, C” in place of ¢,/ and ©’ respectively, and (Aa), (AB), (Ay), (Ad) in as of %, B, ¥ y, and 5 respectively. (A8) being small, and (Az) very small, it is clear that the new ¢ and ¢' series will converge with the necessary rapidity. Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 258 626 Mr. Bernard Cavanagh on Comparison with experimental data me in. ‘ esti- mated” concentrations will thus give (Ax), (AB), (Ay), and (46d), and hence also (Ay') and (Aé') [from (74) ]. This will give a, 2, 7'', and 8", and we can then at ¢ once get ¥ and 6, for it is sisi shown lat 7" =17+B(Aa)} ; ce 5 = {8+ 25(Aa) + BL (AB)?+ (Aa)’]}. . . (75) This procedure being available wens 2 a and # are. large, the highest concentrations, for which four-term expansions of « and @, suffice, fall machin the scope of the results given here and it has been seen that, these concentrations are high. In considering (62) in the sequel, it is understood that this special procedure will be resorted to where necessary, in which case (Aa) etc. will replace a etc. in the eee Hees dasnty SeUCe MAIN RESULT OF THE ANALYSIS. Having now completed the analysi s we may conveniently use Mo, C, cs, in place of My’, C’, ¢;', for the “ experimental ’ i quantities, since the corresponding “true” quantities will but seldom have to be considered. If, then, we write Jog = RO? (t+ 20 + tC? + 140°) JRO +E0 “CLOW (62) takes the form (76) 0 7 OY = byt ROTI + Gye | r (77) oF = 9,— —Rloge+J,+G, j and, in fact, if, corresponding to G, we write b=Gy F 2605, (iol ok bode) ore Jada Pe es) Lae ha aon = —RC?(t, + $40 + 4430? + 44,0*%), . . (80) (77) can be condensed to = O+R3q(1—loga)+I+6, . . (81) which is a the simplest and most concise expression of the main result of the analysis of this and the preceding papers. Molecular Thermodynamics. 627 PRACTICAL SIGNIFICANCE AND APPLICATION. Now the equation HL = $+ RXe(L— log «), BMOD RO} or less concisely, but more practically, (83) is equivalent to the Raoult-van’t Hoff limiting laws of dilute solution, the older criterion of “ perfect’ behaviour. In the circumstances considered in this paper (as stated at the beginning) C and ¢, are “experimental ”—that is, directly determinable—quantities, and to the experimenter, as such, J and G will appear together simply as the measurable departure (J +G) from “perfect” behaviour according to the older (van’t Hoff) criterion. 3 In fact, in terms of the “activity coefficient” (y) of G. N. Lewis, and the osmotic coefficient (l1—g) of Bronsted, expressions pow much used in practice for the observed de- partures from (van’t Hoff’s) “ perfect” behaviour, we have logy= ~7(.+G) oS (84) LU @—-1= But Gy) where @ is a mean quantity for the whole solution, characteristic, therefore, of the given solutes, mixed in given proportions. It may be convenient therefore to call (J+G) the “‘apparent” general terms, G the “true” general terms, and J the “ pseudo-general”’ terms. It is, of course, only G which represents real departure from perfect behaviour (as defined by the linearity ot the full ‘ molecular” expression for % ; see second paper, Section IT.). It is clear that any physical interpretation of the ‘“‘apparent”’ general terms based upon the ignoring of either J or G, without due consideration and adequate grounds, must be unsound, and that in general neither will be negligible, so that a separation of J from G must be attempted. This will only be possible by a critical application of the 28 2 | ; 628 Mr. Bernard Cavanagh on limitations imposed upon J and G by their respective physical significances. It seems reasonable, however, that when all possible has been done in this direction, if doubt still remains as to whether all or some portion of the observed effects should be assigned to J or to G, a “casting-vote ” should be given (temporarily) in favour of J, since J does not re- present real departure from perfect behaviour, properly defined, It is possible that G may be completely determined on theoretical grounds alone in certain cases, the nearest approach to this, so far, being in the case of electrolytes, or rather of a mixture of ions all of the same valency, for which the (probable) form and order of magnitude of G, in (sufficiently) dilute solution, was obtained in the first of these papers, on the basis of a calculation of Milner’s. With regard to J, besides the fact that it takes the form of a series of ascending integral powers of O, the total solute concentration, a good deal can be deduced from the form of the coefficients as t,, etc., and the physical significance of the three series of quantities, the ‘“solvent-constants ” (a;, M2, a3), the a-series (2, 8, F % 5), and the a,-series (cs,, Bsy Ys9 bs) THe Form OF THE COEFFICIENTS ¢,, ¢,', ETC. (59), (61), (63), (64) give us these coefficients as functions, at first sight rather complex, of these three series of quantities, but for practical purposes this complexity is only apparent, for each successive term of the J;-series introduces Just one further term from each of the three series of physically significant quantities, and introduces it in the first degree, while in J,, the same is the case except that the a,-series does not enter. These coefficients can be obtained in a much simpler and more practical form by introducing the quantities fs, hs, ks, ..., and the corresponding mean quantities t, h, k, ..., such that ky—Ys— Lah =h— y = Sayh | LU aa Th Cena RR i (as = — 40; +4a,a,—30,°~ — ya,° J Molecular Thermodynamies. . 629° anlso on, for we then obtain t=t; ty’ =t+t, ) =P +h; ty! = (tty + Bt?) + (h+ Bhs) | re tz=t+3th+h; te’ =(@t, +40) +(2th+ tat ih.) hs ) + (b+ 3A) J and so on. Clearly the coefficients of J,, alone, determined term by term (with increasing concentration) will yield the quantities t, h, etc., one by one, while, if the coefticients of both J,, and 3 are BiGRibd the pairs (¢,, t;'), (ts, &'), ete., will eaatd in succession the pairs of quantities (, bx) yu Rs, Ng )n, CLG It must be observed, however, that while ¢;, h,, etc., are constants peculiar to the solute s (in a given solvent), on the other hand, ¢, h, etc., are the corresponding average quantities for the whole solution, and so depend upon the relative pro- portion in which these solutes are present [see (56a)|. Some cases where t, h, etc., vary under the conditions of experi- ment can in fact be dealt with, without difficulty, as will be shown later, but, of course, it is simpler if they are constant. Two important cases where this is so present themselves, viz., the case where there is only one solute and the case where there are two, but the concentration of the one is negligible in comparison with that of the “second,” the former being the solute s. ‘hese two cases gain further importance from the fact that the form of G, the “true” general terms, will here probably be simplest, depending in the second case almost entirely on the ‘‘ second”’ solute. The last point suggests, however, that in the matter of separating J from G, cases where ¢, h, etc., vary under the conditions of experiment may prove more useful. THE QUANTITIES 255) %, @y, ETC. (85) shows that the evaluation of t, ts, ete., would nee give us these quantities (in general), but since the differences shown in (85) will generally be much smaller than the (corresponding) quantities a, @,,, etc., a rough estimate of these differences (all fairly nearly expressible in terms of a, which is something slightly greater than the mean molecular weight of the pure solvent) will suffice tu determine 2, &,,, etc., with a relative error much less than that of the estimate. Actually, to determine the solvent constants experimentally 630 Mr. Bernard Cavanagh on (and so permit accurate evaluation of @,,, %, ete.) we shall require an entirely unsolvated solute (such as an inert gas, in water), for which t, will be (—4a,), and so on * That only the solute s need be unsolvated is an iaroari point, since the solubility of the more inert gases (for SA is small. PuHysicaL SIGNIFICANCE AND LIMITATIONS OF THE (JUANTITIES, The quantities a, as, 8s, etc., though more difficult to” estimate than the solvent-constants, are not, of course, merely arbitrary parameters by any means (49), for instance, shows that if only one solvate is formed by the solute s, then (.—S)=M, mee ia so that the composition of the solvate is determinable from a,, and 8s, and, of course, M, will have to be an integral number of times the molecular weight of the solvent, Similarly, if two solvates are formed and the amount of (s) remaining unsolvated is very small, then, on the assumption of perfect solution a, 8;,,and ys =reil| ee to determine the composition of both solvates, according to the analysis men- tioned, but not inserted, above. If the unsolvated residue is not negligible, 8, will be required. ae: A more general limitation is that so long as only * ‘positive’ solvation is in question (see first pages of this paper) a, cannot be negative, and ~, cannot be positive, the latter being demonstrable from ‘“ Le Chatelier’s Principle.” In other words, t, may be a fairly large positive quantity, but one a quite ‘small negative quantity, ranging, in fact, from —+a, to the highest probable value for M,. Similarly, hs; can range from (very nearly) j4,a,? to the highest probable value for 1M,?.. When the solvent is water a, may be 60, possibly even higher near 0° C., and M, might easily be 180 (decahydrate), while, of course, hydrates much higher than this are known even in the solid state with low aqueous, vapour-pressure, so that a value of M, as high as 300 would not be improbable in certain cases. : * Compare here 2nd paper loe. crt., p. 241. Molecular Thermodynamics. 631 RANGES oF APPLICABILITY OF APPROXIMATIONS. -Our four-term expansion of a, leads to (62) or (76) or (80) which may be called a “ fifth-approximation” and from which, by curtailing the series one term at a time, the fourth, third, second, and first approximations are obtained, Owing to the diversity of possible magnitudes of the quantities as, %, Qs, etc., precise statement of the ranges of applicability of these successive approximations is not possible, particularly in the case of the higher ones, but some indication of probable range of the lower, for aqueous solution, may be given, The third (as far as ¢,) will commonly suffice as far as Molar (M.) or 2M. solution (but will often go further), the second (as far as ¢,) probably up to $M. or M. (sometimes further), while the first in which the pseudo-general terms J are omitted altogether cannot be assumed to be accurate to - 0-2 per cent. above hundredth-molar (00) concentration. When hydrates higher than the decahydrate are not im- probable, the upper limit for the first approximation must be set still lower. EXAMPLES OF APPLICATION. Detailed application of the results of these papers to existing data must be postponed, but it may be of interest to cite one or two of the simpler instances. Consider, first, a perfect solution ( vanishing) of middle concentration, for which our “ second approximation” will suffice, so that oH gy t ROL +AU). . Se pik) If P be the osmotic pressure, defined and measured so that the pressure on the pure solvent is relatively negligible, and @ the coefficient of compressibility of the pure solvent, then writing (with van’t Hoff) Vo for enn and considering poU the smallness of ¢,C relative to unity, we easily obtain P [vo- 2 ee pART) | are 2 eG 0 ad where, if we can neglect the effect of the pressures used on the solyation of the solute, the quantity in round brackets is a constant at constant temperature. 632 Mr. Bernard Cavanagh on Since, further, at these middle concentrations the density of the solution is expressible as a linear function of the con- centration, so that the ‘“partial-molal volume” » of the solute is a constant (at constant temperature), VeVi oo ee where V is the volume of solution containing one gm.-molecule of solute, and (89) can take the form PIV es eR a where 6 is constant at constant {emperature, and ba [e+ fe Ss etallp Mh i) Equation (91) was put forward by O. Sackur *, in analogy with van der Waal’s equation for imperfect gases, and was shown to represent the data for middle concentrations. We see that for a perfect solute this equation can be predicted for such concentrations. Now consider a slightly soluble solute s in a solution con- taining also a “second” solute in concentration ©, relative to ane? c; is negligible. If G, is negligible, we have oY = $,—Rlog tend... ae In solubility measurement we have the solution in equi- librium with a phase consisting entirely of the solute s at constant temperature and pressure, so that log ¢,— = log cYs)=constant, . . (94) where ¢, is the solubility in presence of the concentration C of the “second” solute. If ¢,, be the solubility in absence of a ‘second ” solute, we have OG 2, = 02 ge as nee recemee | (5) 55) 80 IE or log (= logy.) =— sds. |). ey For concentrations at which the ‘second approximation ” suffices, we expect then to find log =4'O= (¢+4,)C, i eS a where ¢;’ is a constant, since ¢ depends practically entirely * Zeitschr. f. Phys. Chem. (1910). Molecular Thermodynamics. 633 on the “second” solute. We thus obtain a simple first- approximation law for the solubility lowering of wu slightly soluble perfect solute produced by the addition of other solutes. If the “second” solute be an electrolyte G,, will not vanish, but since at the very moderate concentrations to which (97) would apply, G will probably be practically entirely due to the electrostatic forces between the ions, G, may be expected to vanish if (s) is a non-electrolyte. Thus (97) may be expected to apply to the solubility lowering of slightly soluble non-electrolytes—the more inert gases, for instance—produced by the addition of small quantities of salts, providing the complete ionization theory is accepted. (97) therefore constitutes a prediction, compatible with perfect behaviour on the part of the non-electrolyte of the empirical law of Setchenow, which latter should, however, be re-written in terms of solvent-weight (instead of solution- volume) concentrations. For binary electrolytes we see that 1000C (roughly) equals the equivalent or twice the equivalent concentration according as the ions are bivalent or univalent respectively, and that Zi Nie Pee totsy. re 6! @ fee (98) where ¢, is peculiar to the cation, ¢, to the anion, and ¢, to the slightly soluble non-electrolyte. The data tabulated by Rothmund seem on investigation to be of somewhat doabttul accuracy, and the concentrations examined too high. Moreover, they are expressed (as per- centages) in terms of Setchenow’s law, that is, in terms of solution-volume concentrations, and the recalculation is thus necessary. ; They serve at once, however, to show that our prediction is quantitatively quite of the right order of magnitude to suffice. In particular, the smallness of the few negative values should be observed. 3 Among the more reliable figures there is also some evidence of the parallelisms which the additive form of (98) would predict for the recalculated values, while the greater degree of solvation clearly assigned to the ions of higher valency, (Mgtt, SO,--, CO;-~), is quite what we should expect. Closer examination and more accurate measurement will be necessary to determine how far and for what range of con- centration G, can be neglected here. Finally, the single case (as yet) of (probably) accurately known ‘‘ true” general terms may be introduced. 634 Mr. Bernard Cavanagh on It was shown in the first of these papers that in the case of a mixture of ions, all of the same valency, theory alone is able to predict, for (sufficiently) dilute solution, a simple limiting form | G =—R¢'C??,. hie oad ee ee where 20, is the total ion-concentration*. The range of validity of the ‘ point-charge’’ assumption, on which this form depends, should be greatest in the case of the univalent ions of simple structure, and might here extend to, or even above, tenth-normal concentration. Assuming this, and supposing that, while other solutes (typified by s) besides the ions (typified by 7) are present, there are no other “true” general terms at these low eoncentrations, we have ; Nh o+R[C—2Xe, log ¢,— Xe; log ¢;| + J —Rd'C,3?, (100) or 1 eOe @-l=podut2e C= se 1 ae en og Ve ae Ro» 9 ° ° (101) log Wan ea al, ale 3h'CO.'? where 2c; is 2C;, and C is Se,+ Xe;. At the concentrations considered, J, of course, reduces to one term, and we have \ gate page | log ys = (t+t.)C, | 5 uence lly log yj = ¢+1)C + 39'C?”, | , In practice, logy; will always: be a mean quantity for the two ions of the electrolyte J aes so that log yj(=4 log Yje- Yja) = (EF Atje + Etja)C+34'C:", (103) where ¢;, and tj are peculiar to cation and anion respectively. For a single binary electrolyte by itself log yj = 2c + ta) Oi + 36' Ci ie G1 = Whe ttya)Cit 3's? J” * See equations (88), (89), (92) of 1st paper. (104) Molecular Thermodynamics. 635 CO; being 2c;. Now tj and tj, being constants peculiar to the particular ions constituting the. electrolyte, it is plain that we have here a possible explanation of ‘the specific divergences of the simple uni-uni-valent electrolytes at and below (say) tenth-normal concentration, not involving the rejection of the “ point-charge” assumption at these concen- trations. As might be expected, the limiting “law” Uc!” for @ or logy, found by G. N. Lewis to apply, below: (about) hundredth normal solution, to uni-uni-valent electrolytes, ean be extended to cover the data to ligher concentrations in the form (bc?+ ac), where a, unlike 6, is specific, or peculiar, to the particular electrolyte. But, further, the values of a required are quite of the order of magnitude to accord with the interpretation of (104). One other particular case of (103) may be cited, viz. the case of two electrolytes together, the concentration of the one, j, being negligible compared with that of the other, 2, so that C,;is practically 2¢;, and f depending on 2 only can be written (4¢;.+4tia), whence log y; = (tie +tyat ticttia)C; +36'Cit?, . (105) in which, it is seen, the specific properties (as regards solva- tion) of the jourions present enter very simply, sy mmetr ically and additively. ~ These few brief and very*limited illustrations must suffice for the present rather lengthy paper. APPENDIX ].—THE Expansions oF U anp V. - As in the previous treatment of “complex solvents” the full consideration of the expansions of U and V, the total energy and volume of the solution, is postponed, but one point concerning the simple linear forms applying to “ perfect solution” is briefly considered. We have then the “ molecular” expressions cae Oe ee oe ee Us, + Dats,Us, 19 >, - (106) Q = 20,90, + 2Ms[ (1 —Xs) Qo + 291] and, starting from the results of the previous paper, we have U = Moum + duo, Ano, + in,[ (1— X,) us, + Lars,Us, | = My'uy + Su, Ano, + Sn; [ (1 — Xs) us, + Dare,e, “2 ot sUyg | - 636 Mr. Bernard Cavanagh on Writing | Lt l(1—X,)us, + Sats, — asm | = Us, « (107) C>0 7 and, of course, PNT \ 9 ON i we have \ 0s) U = Mo’ um t+ Sngtte + {So Ano, + 75d (ls, — Us, — Mewar) Axs,}, (108) and, similarly, V = My om + angus + {Sw,Ano, + rsd (Vs, — Vs — Ms,vm) Avs}; ; (109) Q = Moau t+ Srgs + {Eo Ano, + 2Ns=(ds,— Io — Ms, qu) Ate} (110) It is plain (cf simple demonstration in previous paper) that the bracketed quantity in (110) is the heat developed on “infinite” dilution of the solution, and it also clearly represents the heat of the chemical action involved in the change of the free solvent and of the solvation equilibria to their limiting states pertaining to “infinite” dilution. When only one solute, (s), is present, this heat of dilution assumes the simple form—per gm. molecule of solute— [| SaoAno + 2(Gs— qu Magn) Ae een. Ok Similarly, the bracketed quantity in (109) is the “contraction of dilution.” ) We saw in the previous paper that \ uM = TS, OM; (112) um =—T 0 out Op The demonstration that a hg == ae a. auees (113) ; ey Us mer ee Molecular Thermodynamics. 637 is not quite so obvious, but will be obtained without difficulty if (13), (10), (6), and (3) are used in the order named. Finally, we have the interesting relations 1? ea se {uy Acy,’ Lg ce,’ > (u,, .e Us, Ate M,%) Aes | oe GOES _ ay Op se {Ziv Ay,’ a XC, (v,, ik, ur Yay M, vy) on J | APPENDIX II. The practical significance of (39) may be illustrated in a simple way, by means of a suppositious case in connexion with strong electrolytes, entirely analogous to that given in the appendix to the previous (second) paper. The ‘general terms” for a dilute solution of a binary strong electrolyte being M,'G'"'= RM)'¢'c,'2?, - . iy ce wie Gena) we suppose that the “effective” dielectric constant (D) depends on ¢,' in such a way that @¢’ is a linear function of ¢,', i Sela ag opis nsw C16) If this effect of c,’ on D is independent of its effect on the solvation-equilibria, then we have G!! sp Oa 0, so that eS Ge ee a. lean Gee. eho 62S 3Rdo'e./ (1+ ac,'), §. (117) Gar= Gye" = —ER¢'e,! 92 = —ARGole,!3°U + a0!) { If, on the other hand, the effect of ¢,’ on D is entirely dependent on its effect on the solvation-equilibrium, then OG da, 04s, deg! fs — Re, 3/2 o¢ og = Rady'e,’9? 638 Mr. Bernard Cavanagh on ~ and OG" | ee 5G" Oe lie 0 Cs 1526"4, te eae = 2Rado ¢s. ae so that eee \ 7 da, =Rey'e (1+ face’) C=0 Cs" ; : | G; = G,!’ a fr ae ke | oles = 3Rdolcs. Het a5 Bucs’) Gy Gy" re moun ‘es. 3/2(1 + ac) J (118) - Asin the previous paper, it is to be remarked that in such a case, if D were the ordinary .measurable or “ bulk ” dielectric constant, the necessary modification of the general terms (if not negligible) might be introduced without know- ledge of the way in which the degree of solvation of the ‘solute was able to affect D. In both cases also it may be observed that something might be learned by the introduction of other solutes, since both the constitution of the solvent and degree of solvation depend mainly, if not entirely, on the total concentration ‘C’—not on the concentration c,’ of the solute (e ie) considered. The case of electrolytes provides an interesting and important example of the (probable) fulfilment of the con- dition for the vanishing of (40), though the solution is not perfect. It seems quite likely that—at least in the more dilute solutions—the ions will behave essentially and approxi- mately as point-charges, whether solvated or not. The general terms for a solution of ions obtained in the last section of the first of these papers would then hold equally for solvated and unsolvated ions. That is, we should have i ! i G =. = Sa ee ey and since M;Gy;’ will be a much smaller quantity in dilute solution, Gra," ', ete., will probably be quite small. For Gz; ete., ie vanish altogether, we should require that the “ bound” or “absorbed” solvent should continue to function in the same way as the free solvent, as separating dielectric between the point-charges. Molecular Thermodynamecs. 639 APPENDIX III. The form of the cofficients in (86) suggests an alternative method of meeting the practical difficulty of slow convergence of the J series, which will be preferable in many cases ‘though not always) to that already given on account of its very attractive simplicity—both formal and practical. Extending (85) to the fourth term (1, —8,—a,ks+ hagh,) =((—8 —a,k + Ja,h) =(— yeast + 4aj7a, —taya3— fay’ +a.) ~ Jgay*, . (119) we have for the coefficients of the Jy,-series, t; tz=(t?+ 3th+k ain aay aga sbrs Meat t ee and on comparing this with (59) it is at once seen that if 1 . we write CU’ for (© a pat) then agiu= fa —1)=0' +f +20" +10". bah) Also (but less obviously), if Ene, a Os we find Rlog¢—J,=Kilog¢, —J;, .,: + (122) where —J, =RO (Gs +h + KC? +1,0%),.. .° .(1238) These ‘“ concentrations ” (c;', C’) have no simple physical significance, though they will approximate roughly to the “true” concentrations of the earlier analysis. They are, however, a great practical convenience, since the series in (121) and (123) will clearly converge “ practically”? in few terms so long as the expansions of a, and @ (in ascending integral powers of the total “true” concentration) do so, that is, as we have seen, up to very high concentrations. Now, supposing either that we can neglect G (thus making, tentatively, the assumption of perfect solution), or that we can separate G from J in our measurements, then solvent-separation data will give us Jy and solute-separation 640 On Molecular Thermodynamics. | data will determine (differences in) Rlogc,—J,, that is, if Rlogc,'—Js', In the former, therefore, = as well as C is a measured quantity, and when obtained (graphically) as a series of integral powers of C' gives the quantities ¢, h, ..., directly. | But without (and independently of the precision of) this (ie Ae rains: C 3 determination of ¢t, h, .,., the measurement of cq makes ¢,! a measured quantity, so that a combination of solvent-separa- tion data and solute-separation data gives us J,' directly as a measured quantity, and this when obtained (graphically) as a series of integral powers of O’ gives us ¢,, hs, -.., referring to the solute “separated,” at once. It is, of course, in the obtaining of the series to express, as accurately as possible, the data, that the essential ad- vantage, the rapid convergency, attained by the use of the ‘¢ convenience-concentrations ” (c,', C’), appears. The sim- plification, however, of the coefficients of the series (as compared with (62)) will probably mean greater accuracy in the evaluation of t, ME Oe ipeeal Hee SumMARY OF Papers II, anp ITI, Paper II., Section 1. A postulate, a rider to the definition of “a molecule,” is proposed as the basis of rigorous molecular thermodynamics, and its significance is discussed, Section 11. The theoretical basis of the method and the nature of the general problem are outlined. Section 1. Analysis of the problem of ‘complex sol-: vemlise yy | Paper III. (the present) contains some analysis of the problem of “solvation of solutes,” to which is attached a discussion of the results with some brief preliminary illustrations. Balliol College, March 1922, [64a J LVI. The Law of Distribution of Particles in Colloidal Suspensions, with Special Reference to Perrin’s Investi- gations. By AuFreD W. Porter, D.Sc., F.R.S., FInst.P., President of the Faraday Society, and J. J. Hepass, B.Sc. * iO the Proceedings of the Royal Society for January last (A, vol. 100, No. 705) KE. F. Burton criticises Perrin’s work on the law of distribution of particles in colloidal solutions. In the course of his work Perrin shows that, at any rate when the solution is dilute, we should expect a law of distribution of the particles analogous to that which characterises the distribution of the molecules in an atmosphere. By balancing the osmotic pressure of the particles against the effect of gravity he deduces the equation RT n Ty loge No = V (pi—ps)g(h— ho), where n and zy are the numbers of particles present per unit volume (i.e., the numerical concentration) at depths h and ho, V is the volume of each particle, p, and py, the densities of the material of the particles and of the medium respectively. Experimentally, for example, Perrin obtains for particles of gamboge, 2°12 x 10~° em. diameter, the number of particles at four depths differing successively by 3x 1073 cm. and finds values proportional to the numbers 12, 22-6, 47, and 100, which numbers are almost in the same proportion as the members of the geometric progression 11'1, 23, 48, 100. Burton’s criticism amounts to saying that if this law of doubling continued as the depth increases then at 3 cms. depth the concentration should become 21°” times the: first value given (viz., 12). Now this ratio isa number containing more than 300 digits; and even casual observation shows that no such increase occurs. In faet, Burton makes experi- ments to show that throughout most of a tall column of suspensoid there is no sensible change in the concentration. Perrin’s own observations were confined to very small ranges near the top of the suspensoid where the concentration was very small, in order that his theoretical and experimental work might correspond to one another. He says in “ Les Atémes,” § 61, “Ce n’est pas sur une hauteur de quelques centimetres ou méme de quelques millimétres, mais sur des * Communicated by the Authors. A paper read before the Faraday Society, Monday, June 26th, 1922. Phil. Mag. 8. 6. Vol. 44. No. 261. Sept. 1922. 2T 642 Prof. Porter and Mr. Hedges on the Law of hauteurs inférieures au dixieme de millimetre, que l’on peut étudier utilement les émulsions que j’ai employées.” We have thought it important not only to prove that the concentration tends to uniformity as the depth increases, but also to find the law of change when the concentration ceases to be sufficiently dilute for the perfect gas equations to hold good. Haperimental, A suspension of gamboge was prepared by rubbing the solid under distilled water with a soit brush. From this, one in which the particles were practically of the same size was obtained by means of fractional centrifuging. The process actually adopted was that worked out and described by H. Talbot Paris *. To count the number of particles at various depths, a modified arrangement of the Zeiss ultra-microscope was used. The cell in which the susnensoid was contained was viewed by the microscope placed with its axis horizontal, while it was illuminated by a vertical beam passing down- wards through the cover slip which served as a lid to the cell. This vertical beam was obtained from the horizontal beam passing from the lantern through the usual lens system by means of suitably placed reflexion prisms. The final condensing lens with its screw adjustment was placed with its axis vertical, and the cell was attached to and movable with this lens, so that various depths could be examined in turn. An iris diaphragm in the eyepiece per- mitted the field to be cut down until the number of particles visible at any moment was sufficiently small to be estimated at a glance. ‘lo diminish convective effects the light was passed through a water-cell, and a shutter was employed so that the light only passed through the observation-cell for a time sufficient to enabie the particles to be counted. With the exterior of the cell blackened and all stray light screened off, it was found possible to count the particles in the field of view over a range of several millimetres from the surface. This arrangement was not convenient for making obser-. vations so close to the surface as those made by Perrin which were all confined within the range of less than Q:1 mm. ; but this was no drawback to the object we had in view. The method of obtaining the concentration at any depth was to count the number of particles in the field of view from twenty to forty times at intervals of ten seconds, and * Phil. Mag. xxx., October 1915. Distribution of Particles in Colloidal Suspensions. 643 then to take the mean of these counts. The volume to which these counts correspond is governed by the diameter of the circular field and by the depth of focus of the objective, both of which were determined ; its value was 2°:1x107‘ cm.?*, ‘The depths were measured (by means of the micrometer screw which moved the cell) from an arbitrary level, which was found to be about ‘023 cm. from the surface. The depth reckoned trom the surface we denote by y, the number -of particles per unit volume (7. e., the numerical concentration ) by n. Several complete sets of observations were taken in order to practice the method. The results for the final sets are given below :— y. -023.|-033.|-043.|-063.| -083.|-103./-123,/-143. Number of counts ............ ee 20 20 | 20. 20 40— “20. 20 “40° Total number of parlinles cacaecd a % 1 s- 30 72 | 40. AL 82 Number per citi Ie eee ons. ol "20 “45° ‘75 1:50 [1-825 2°00 |2:05 2:05 MXIOM6 oo ceveeesesessee veseeeees| ‘9B [214 8-61 [714 869 [952 9-76 [9-76 Professor Perrin’s calculations are explicitly based upon the assumption of the application of the laws of perfect gases to dilute solutions. It is easy formally to extend them to solutions of any concentration. This was done by one of us for true solutions nie COLT *. Solufion Solvent” Imagine a column of solution to be put into connexion ‘with a column of pure solvent at two points through semi- permeable membranes, the difference of depths of these points * Porter, Faraday Society. Discussion on Osmostic Pressure, 1917. 2T 2 644 Prof. Porter and Mr. Hedges on the Law of being dy (reckoned positive when downwards). The concen- tration of the solution at any depth is c where c=nm, m the mass of a particle and n the number of particles per c.c. (2. é., the numerical concentration). Considering the osmotic pressure P, 2.¢. p—po, as being a function both of the hydro- static pressure of the solution, p, and of the numerical concentration, we have the mathematical identity rapt ea dp \Op Jn” \On/p dy dp’ Now eo —“—* where s is the shrinkage*, and simple uM a utichale dpy u-—o hydrostatic considerations give — =1l———-= ; also: dp dp dip U ae 3 Therefore Sve OF an oF pe ae This formula is exact, and is independent of any particular hypothesis of the mechanism of the variation. Now it was shown by Sackur and by one of us that the variation of P with concentration in the case of a sugar solution can be represented very nearly, up to high concen- trations, by the formula nrT = (1—bn)’ where 6 is a constant which is of the same order of size as, but is larger than, a molecule of sugar. If we assume the applicability of the same type of formula to a suspension of gamboge : OF organ et on (1—bn)?° In these formule 7, which applies to an actual particle, is. connected with the usual molecular gas constant, Ry (which refers to one gram-molecule), by the equation R : t= nN? where N is Avogadro’s number. dn __s—oa gN Hence dy = “uo Rol (1—bn)?. * Porter, Proc. Roy. Soc., A. 1907, p. 522. Distribution of Particles in Colloidal Suspensions. 645 If we assume that there is no contraction when gamboge and water are intermingled o=cy+(l—c)u and s=u, where y=density of solid gamboge. Hence o—s=c(y—u) =nm(y—u). Further writing K= a ” and, putting o in denominator as equal to u [which is justi- fiable, because even the strongest suspensions of gamboge are fairly dilute], dn they, Pe = Kn(1—én)?. The integral of this equation is n i CS spear uty aia maT where A is the constant of integration which can be expressed in terms of (the unknown) concentration when y=0. This is a curve which tends asymptotically for large values of v to the value n, =1/b; and which has a point of inflexion for a. |. A ais of this kind can be fitted to the experimental curve within the limits of experimental accuracy (fig. 2). The following values are obtained by taking b=10°2 4% 105" em:-:and:-K = 1291. y (in ems.)...... 024-082, 0375: 047 “061 089 ‘115 ca A Tae 1 2 3 5 7 9 9:7 A closer fit could, of course, be obtained by allowing 6 to vary with the concentration, as was done in examining the osmotic pressure of sugar solutions*. It is difficult, however, in the present problem, to give anything but an empirical significance to this constant. It enters into the osmotic pressure in the same way as the least volume of the liquid enters into the gas equation. But in this case it would mean that even in the fairly dilute concentrated suspension of gamboge the effective volume of the particles is the volume of the solution itself—that is to say, that the suspensoid * Porter, loc. cit. tame ws —— 646 Prof. Porter and Mr. Hedges on the Law of plus the atmosphere (or ocean) of solvent (or second phase) which surrounded it, and to which it was attached (so that motion of one entailed similar motion of the other), constituted the whole of the substance present. The radius calculated by considering b ag the effective volume is 28x 10-+cm. Now it is hard to think that the whole of the liquid in the domain of a particle should be so attached to the particle as to form practically part of it, for, if this were so, the whole suspension would be rigid. It might be conceivable in the case of even a dilute concentration of gelatine, for this can form a rigid gel; no such rigidity is observed in the case of gamboge. Itis noteworthy, however, that if 6 be regarded as an effective volume, then b and K. are not independent constants, for K is proportional to the effective volume of the particle. When the radius is calcu- lated from K, it is found to be 1:6 x 10~° em., which is very much less than when calculated from 6, but is certainly a measure of the effective radius of the particles that were examined. We return to this point later. | It is safer, however, to regard 6 as merely an empirical constant. It may be pointed out that Callendar’s. equation for unsaturated steam is of the form p(w—b) =RT, where, however, b is even negative and represents the com- bined effects arising from the finite size of the molecules and the attractive forces which tend to “ co-aggregate” the molecules. It is noteworthy, however, that, if 5 1s to become large and remain positive, repulsive forces are required. Burton, in the second edition of his monograph on the physical properties of colloidal solutions, page 87, attempts to explain the present problem by considering repulsion arising from the electrical charges in the particles. He concludes that the forces on any layer due to the rest of the solution will be of the form kne per unit charge on the layer dh, where k, he considers, may be taken as a constant, at any rate for regions near the surface, and e is the charge in electrostatic units on each particle. Consequently the total force per unit area is kn7e*"dh. A term of this kind is taken to represent the excess electrical repulsion of all the particles below dh over that due to those above dh. Perrin’s equation then becomes, for the forces balancing per unit area, : = dn + kn*e?dh=nV (d —w) gdh. Mistribution of Particles in Colloidal Suspensions. 647 Writing this in the form Adn=n(C—Bn)dh, and solving, he gets C = B+ Ke- Oa? in which K=(C— Bno)/np and mp is the concentration when h=0. To obtain Perrin’s formula B must be put equal to zero. ‘The ratio C/A can therefore be obtained from Perrin’s experiments. Calculation shows that e~/4 tends rapidly to zero.as h increases, and ultimately n becomes YON (G—w)¢ rT a) 2 es) ke? The depth at which this uniform concentration is practi- cally attained will depend upon the relative values of K and B; it will be nearer the surface the larger the electrical forces are compared with the gravitational. Now there are serious objections to the theory as thus stated. In the first place, if the particles really contained charges all of one sign only they would tend to move toward the boundary. This is the equivalent of the fundamental electrical fact that statical charges reside close to the surface of conductors. When we are dealing with large particles instead of electrons, there is no doubt that they would occupy a larger region, instead of a thin superficial area, but still there would be an accumulation at the boundary. This is the opposite to what is observed. But the charges in the solution are not only ef one sign. The solution, as a whole, is uncharged; consequently an equal opposite charge is to be looked for. This opposite charge is the second member of the double layer close to the surface of each particle. When the existence of this double layer is recognized, the electric forces between the particles become zero, except in so far as relative displacement takes place by induction between two members of a layer so as to give it an electrical movement. In this case the force between two such doublets in the equilibrium state will, on the average, be an attraction and not a repulsion. Now we have fitted a curve calculated from Burton's equation to the experimental points. ‘hey are shown by large circles on the figure and are seen fo fit the experiments remarkably well. In view of the above objections to the 648 Prof. Porter and Mr. Hedges on the Law of theory, this cannot be taken as indicating more than that in the differential equation for the distribution there is a term. proportional to n? in addition to the one depending on n. Digeez ee ees Soe. 7 (stern Mn AGRE REN ERORROS DS See ORe USOT TEE Na SIC ca aa el A LL ene no L OOo 87 eet eal htt al 7) SE Caen eae eee ee eT 970 ae a oa BECO R Ue ee ee a oe Sem Te ea VST eal A ig SAREE See | > Gennes Zee 7 CRE a se eee eee = A ae 1 fT Ae ea aa a wae im me wae eee ares eae Bee cane eee eee eee Ltt | ee eel Cr ‘Ol -02 :03 :04 -05 :06 -07 :08 -09 -10 oy ea ee The limiting volume of n is still determined, not by a, but by 6, and this is so whether a is positive or negative. Further light is thrown upon the phenomenon by taking ‘suspensions of various limiting concentrations. In fig. 3 are shown experimental curves for two cases; in curve 2 the limiting concentration (?.¢., the concentration in all except the upper layers) is twice the value for curve 1. Hence, according to the equation given, the value of b in the latter (i. e., the dilute solution) is twice the value it has for the stronger solution. ‘lhis, again, is quite in accordance with the behaviour of sugar, for which substance more molecules of water were found to be attached to each molecule of sugar in the case of a dilute solution than in the case of a strong one. * Porter, loc. cit. 650 Distribution of Particles in Colloidal Suspensions. These last experiments show that care should be taken to: avoid thinking of the limiting concentration as necessarily giving a saturation value. The same kind of curve is- obtained when more dilute suspensions are investigated. The value of K is also different for the two curves. For curve 1 (fig. 3) it is about 29, while for curve 2 itis 55.. It would seem ion the size of the solid nucleus diminishes to one-half when the suspension is reduced to half strength. Fig. 3. if BEERS Bama suseeccaneeeee HE —p tty me Pea Vane Fai RY aanes en la faves ABH axe vi 2a) ela aie aE a Although Burton’s spivabilbay2 assumption of the existence: of an electrical repulsion between the particles has been here criticised, it might still seem an open question as to whether some other force between them might not be the operative: cause. The chief difficulty applicabie to all possible laws of force is that such forces will be exerted in all directions and must act outwards at every surface. Hence, though near the top of the fluid they might be assumed of such a charac ter as to oppose gravitation, at the bottom they should assist it in producing a change of concentration, which might be expected to be exceedingly large in consequence. Again, at the sides where the forces would act laterally they would produce a change of concentration independent of gravi- | tation. What really happens can be seen from fig. 4,. which represents the approximate ‘relative values of the numerical concentrations actually observed at various points of a cell 5 cm. deep and} cm. broad. It is obvious from the . numbers given in the figure that the only variation of concentration is near the top. It ought to be mentioned that,. since the suspension was covered with a microscope cover- glass, the suspension was surrounded by glass on all sides ;. the peculiar effect at the top is therefore not due to a transition from liquid to air. It would appear from these preliminary determinations that there are in reality three regions in a suspension to- Emission Spectrum of Monatomic Iodine Vapour. 651 which particular names might be given. In the first place, there is a very thin layer close to any surface in which there may be a special value for the mean concentration. This is the layer discussed by Willard Gibbs, and has been called after him. Its thickness is of the order of the range of Fig. 4. intermolecular forces. Secondly, there is the layer studied by Perrin, “inférieure au dixieme de millimetre,” in which the change of concentration of a suspensoid can be calculated from an application of the laws of gases in an analogous way to that in which the change of pressure of an atmosphere is. calculated. This is the Perrin layer. Thirdly, there is a layer of one or two millimetres’ thickness (in the particular cases studied in this paper), in which a further change of concentration occurs which cannot be calculated in the way adopted by Perrin. This gradually merges into the main body of the suspension, throughout which the concentration is: sensibly uniform. LVIL. Onthe Emission Spectrum of Monatomic Iodine Vapour... By Sr. Laypav-Zisemecni, M.Sc., Lecturer in Physics, High School of Agriculture, Warsaw”. I. The aim of this work. . M ANY of the phenomena of the multiple spectra were ascribed to the dissociation of the molecules or even of the atoms (the work of Lockyer may be only mentioned), but there does not seem to be direct experimental evidence for such a theory. On the other hand, the models of atoms and molecules, now generally adopted, lead us to the idea that an atom should give a spectrum quite different from the * Communicated by the Author. Presented to the Polish Academy of Sciences by Prof. L. Natanson, March 7th, 1921, and published in the Bulletin de [Académie Polonaise, série A, 1921. Since that time the experiments have been verified by the Author. 652 Mr. St. Landau-Ziemecki on the Emission spectrum of the molecule. To verify it experimentally the writer chose iodine, the dissociation of which has been well studied and which dissociates at a relatively low temperature. Il. The Experiments. Bodenstein and Starck * have shown that the dissociation equilibrium can be expressed by the formule : py K (C2 ae : We per log K= — pg +1 75 log T—0:000416 T+ 0°422, where p,; is the pressure of the monatomic iodine vapour, pe the pressure of the diatomic vapour, K the equilibrium constant, T the absolute temperature. This expression agrees well with the experiments. Applying that formula, we obtain the following table for the degree of dissociation—that is, the ratio of dissociated molecules to the total number of the molecules which would be present were there not any dis- sociation. Degree of dissociation as a function of pressure and temperature. t=500°C. 600°. 700°. 800°. 900°. 9608: p=imm. 01 0:36 0-76 0:95 0:99 1 ole 0:05 0°19 0°52 0°83 0:94 if f= 10) 5; 0°02 0:06 0:19 0°42 0°68 0-81 Taking these data as the starting point for my experiments, I prepared a Geissler tube from quartz of the form shown ray siaes Ihr. The iodine crystals were at C, and that part of the tube was immersed in a water-bath, the temperature of which varied in the different experiments from 15°-20° C.; thus the corresponding vapour-pressure was always less than 0°25 mm. of mercury (for 20°C.). The central part of the Geissler tube was placed in an electric oven, and the temperature raised from 960°-1000° C. It is seen from the above table that in these conditions the dissociation was fairly complete. * Zeitschrift fiir Elektrochemie, vol. xvi. p. 961 (1910). + Some months after this work was published in the Bulletin of the Pol. Acad., W. Steubing described in the Physikalische Zeitschr. (1921, p. 507) his experiments touching the influence of the temperature on the band spectrum of iodine. The experimental arrangement of this author was similar to mine. Glass tubes were used by him, and thus he was unable to go further than 450° C., when practically there is no possibility of observing any trace of the line spectrum of monatomic iodine, the percentage of dissociated molecules being quite insignificant. The ex- ‘periments of W. Steubing gave interesting results touching the weakening of the band spectrum with the rise of the temperature. His observations are limited to the visible part of the spectrum. This part of the Spectrum was inaccessible to me in the experiments touching the spectrum of monatomic iodine on account of the radiation of the electric oven. Spectrum of Monatomie Iodine Vapour. 653. It is obvious that in the case considered there exists a circu- lation of the vapour, but, for the diameter of the joining quartz tubes, no serious pressure differences could exist ; besides, it was shown by Nernst * and his collaborators that the dissociation formule can be applied to a gas flowing through a pipe, if the flow is not too quick a one. Fig. 1. Two main iodine spectra are known—a band spectrum and _ a line spectrum. The most complete study of these spectra was made by H. Konenf. Condensed, strong discharges in narrow parts of the Geissler tubes give generally the line spectrum ; with wide tubes and feeble discharges we obtain the band spectrum. The first experiments to show the in- fluence of the dissociation on the emission spectrum were made by the author, using Geissler tubes of ordinary shape with vertical capillary tube. No interesting results were obtained. A new spectrum was observed when using a tube “end on” with large central part, as is seen in fig. 1; the central part B had 2 length of about 12 cm. and a diameter of 1 cm.; the parts A, covered with tinfoil, had a diameter of about 3 cm.and a length of about 8 cm. This tube, not * W. Nernst, Theoretische Chemie, 7th ed. p. 709. W. Nernst and H. vy. Wartenberg, Zeitschrift f. physik. Chemie, vol. lvi. p. 535 (1906). + Annalen der Physik, vol. lxy. p. 265 (1898). r654 Mr. St. Landau-Ziemecki on the Emission heated, showed the ordinary band spectrum. The temperature being raised to 960°-1000° C., the conditions of the electric excitation remaining the same, a new line spectrum replaced the band spectrum. The appearance of that spectrum is quite different from the appearance of the spectrum obtained in a narrow tube with condensed discharges. It consists of relatively few lines; two lines in the ultra-violet are especially strong and characteristic ; the others are faint. Some details of the experiments may now be given. The iodine was introduced into the tube by sublimation in the following way. Pure iodine of commerce (Kahlbaum) was re-sublimated in vacuo; the crystals were then put into the glass tube R, (fig. 2), which was to the right connected with ‘the mercury Gaede pump and to the left with the quartz apparatus. Quartz and glass were joined together with Fig. 2. ground : surfaces ; sealing-wax made that joint perfectly ‘tight (W, fig. 2): The apparatus was exhausted during two hours. During that time the U-tube was immersed in solid CO, to prevent the iodine vapour from penetrating into the pump; at the same time the quartz part of the apparatus, ~which was previously chemically cleaned, was now heated in ‘the most energetic manner with a Bunsen burner and finally with a blow-pipe. The apparatus was then cut away from the pump at P,, and the iodine sublimated from R, to Ry, ‘this part being immersed in liquid air. After some hours, R; was in turn immersed in liquid air, and a crust of iodine crystals was formed there. Hvery precaution against con- tamination having been taken, the quartz part of the apparatus was again put in communication with the pump, and after an exhaustion of half-an-hour’s duration the quartz :apparatus with the iodine crystals at R; was cut away from — the pump at P,. The Geissler tube obtained was put in the electric oven shown in fig. 1, and finally bent with the coal- gas oxygen flame in the desired manner. The tube was provided with external electrodes formed by tinfoil cemented to the quartz with a mixture of graphite powder and water. Spectrum of Monatonuc Lodine Vapour. 655 The luminescence of the tube was excited either by con- necting it directly with the secondary of a Ruhmkorff induction coil (15 em. spark), or by using in connexion with the induction coil a high-frequency transformer of the Oudin type, and joining the tube to the circuit as shown schematically in fig. 3; in the latter case a Wehnelt interruptor was used. Fig. 5. The temperature of the electric oven was measured with a ‘thermo-couple Pt—Pt/Rh. By means of two quartz lenses (L in fig. 1) the light was concentrated on the slides of a -quartz spectrograph. At low temperatures the tube has shown a band spectrum. H. Konen * finds that the band spectrum is almost continuous in the ultra-violet part, having only few diffused maxima at A= 3300, 4270, 4520, 4760 A. On the contrary, I found that this band spectrum has a very regular structure with many well-defined maxima. There seems to be no discrepancy between our results: this author has taken photographs with long exposures, and in these circumstances, as I find, all details disappear; on the contrary, my exposures never lasted longer than a few minutes. For a more detailed ‘study of the band spectrum, I used a special quartz tube with external electrodes, having a capillary part in the middle. This tube being connected with the secondary of a Ruhmkorff coil, I found distinct maxima at ~\=422, 404, a99, 301, 300, 3/3, 368, 363, 340, 325, 323, 320; 319, 312, 310, 306, 304, 302, 299 uu; for shorter wave-lengths the maxima were too feeble to be measured. Raising the temperature of the oven, I found that the transformation of the band spectrum into a line spectrum occurred gradually. At550° C. we have yet a band spectrum; * Loc. cit, 696. Emission Spectrum of Monatomic Lodine Vapour. at 650° C. one of the two characteristic lines 3281 A appeared ; the second strong line 3384 A appeared later; at 800°C. it was yet faint. ‘The manner of producing the spectra seemed to be of no influence: I observed the same phenomena whether applying uncondensed discharges of a Ruhmkorft coil provided with the hammer interruptor, or with the Oudin high-frequency transformer, AGUS by an electrolytic: interruptor. I determined the wave-lengths of the main lines of the spectrum observed; they were found to be 4868, .4769, 4680, 4324, 4132, 4100, 3384, 3281, 3081, 2879, 2593, 2583, 2566 A. My spectrograph had a dispersion insufficient for accurate work. The error of the determinations should not exceed 2- 3A for the fainter lines, and 1A for the two stronger ones™ It would Ihe difficult to tell whether all these lines are absolutely new, or whether some of them can be found in the ordinary line spectrum of iodine: the exactness of my measurements is insufficient, taking into account that the lines of iodine are exceedingly numerous; happily there are no lines of the ordinary spectrum in the proximity of 3281 A; that characteristic line is thus certainly a new one. Ill. Results. A new line spectrum of iodine was observed, and wave- lengths of the main lines approximately determined. This. spectrum belongs evidently to the atom of iodine ; it appeared more and more distinctly with increase of the dissociation of iodine molecules; at the same time the band spectrum disappeared. The band spectrum of iodine seems thus to be inherent in the molecule. It was found that the band spectrum possessed in the ultra-violet a very regular structure,. contrary to that which had hitherto been imagined. It may be permitted to the Author to express here his best thanks to Prof. 8. Pienkowski, who in a most generous way has supplied him with a quartz spectrograph. A great part of the expense was covered by the Mianowski Foundation. Warsaw, Physical Laboratory of the State Technical School, founded by H. Wawelberg and 8S. Retwand. May 1922. * In the violet part of the spectrum much greater errors are possible, the dispersion of the quartz prisms being there very small, While this paper was in the press some other lines were measured and the following wave-lengths found: 4640, 4414, 4217, 38576. 2528, 2524, 2516, 2507. Barton & BROWNING. Phil. Mag. Ser. 6, Vol. 44. Pl. TIT. SSS Barron & BROWNING. Phil. Mag. Ser. 6, Vol. 44, Pl. IV. ) i D0 i Phe, wn ah Ve fy, THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCI ENG Eis “A / > ne Qi L ) 19 1 [SIXTH ona Ceri sous NS ny, OCTOBER 1922.“~~ LVIII. Further Studies on the Electron Theory of Solids. The Compressibilities of a Divalent Metal and of the Diamond. Electric and Thermal Conductivities of Metals. By Sir J. J. THomson, O.M., F.RS.* - a paper published in the Philosophical Magazine, April 1922, I calculated on the Electron Theory of Solids the compressibility of monovalent and trivalent elements crystallizing in the regular system. In this paper I propose to do the same fora divalent element. The simplest case is that of calcium, for Hull has shown that it crystallizes in the regular system and that the atoms are arranged in face-centred cubes. Thus, as far as the atoms are concerned, the metal may be supposed to be built up of cubical units, each unit having 4 of an atom at each corner and 4 of an atom at the centré of each of its six faces. Thus each unit contains four atoms; and as calcium is a divalent element, there must be twice as many disposable electrons in the unit as there are atoms, so that each unit must contain 8 electrons. A symmetrical way of arranging these 8 electrons is to put 4 of an electron at the middle point of each side of the unit cube, one electron at the centre of this cube, and one electron at the centres of 4 out of the 8 small cubes into which the unit cube is divided by planes bisecting its sides at right angles. These 4 cubes are chosen so that if we * Communicated by the Author. Phil. Mag. 8. 6. Vol. 44. No. 262. Oct. 1922. 2U 658 Sir J. J . Thomson: Murther Studies on - move parallel to any side of the unit cube, the empty cubes and those containing eJectrons occur alternately. Assuming this to be the constitution of the metal, we can easily calculate the electrostatic potential energy by the method given in the paper referred to. Let H be the charge on the calcium atom, e the charge on an electron, and 2d the side of the cube taken asthe unit. Then the electrostatic potential energy for a single atom 1s sB(-37 +35), where 7 is the distance of an atom and 7’ that of an electron from an atom under consideration. The potential energy of an electron is e si s e heat > Rai} where 7’ is the distance of an electron from the one under consideration. By the method described in the former paper, I find for the electrostatic potential energy of an atom the expression 16°23 36°85 Lehi — oti are which, since H=2e, is equal to e” 7 4, The potential energy of a single electron, if it is one at the middle point of a side of the unit cube or at its centre, I find to be H.———e.-4-)= 5, 83, d while if the electron is one of those at the centre of the - small cubes, the potential energy is ° O 2 5 ye(B. A We 2) = 5 59°65, Since the neutral calcium atom consists of one positive nucleus and two electrons, one of each type, the potential energy per normal atom will be Fr Le 97 Seay het 9 2 = (444 1°325+°415) = 7 6°15. * the Electron Theory of Solids. 659 If A is the density of calcium, M the mass of an atom, since our unit cube with side 2d contains 4 atoms, 4M 8d* 1 =o 1) ae o ‘ Hence the electrostatic potential energy per normal atom is 2a [A \¥8 2 ° 1/3 a 2.615 2 (sr) and the electrostatic potential energy per unit volume is #615 B8(5) M A 4/3 ee mat | =e.715x (i) Hence, by page 736 of the former paper, & the bulk modulus for calcium will be given by FTP bay cn Ex ae 9 (iu) For calcium | ed ed M=40 x 1°64x 107%, This makes 1// the compressibility equal to 5705< Lvs the value found by Richards is 5°5x107, so that the agreement between the calculated and observed values is again quite close. The other divalent metals Mg, Zn, Cd crystallize in the hexagonal system; in this system the elastic properties vary in different directions, and a uniform pressure would produce a change in shape as well as in volume. The arrangement of electrons and atoms appropriate to this case is when the electrons are at the corners of a hexagonal prism and the atom at the centre. To fix the shape and size of the prism we require two lengths, the radius of the base and the height of the prism instead of the one which sufficed for crystals in the regular system. The necessity for taking two variables instead of one makes the calculations more lengthy than those for the regular system, and I shall defer their consideration for the present. 2U 2 A, 660 Sir J. J. Thomson: Further Studies on Compressibility of the Diamond. In the diamond we have a quadrivalent element orale lizing in the regular system. The arrangement of the carbon atom in the diamond has been shown by Sir W. H. Bragg and Professor W. L. Bragg to be given by the following ~ scheme. They occupy a, the corners of a cube; b, the centres of its faces : c, 4 of the centres of the 8 ies into which the large cube is divided by planes bisecting its sides at right angles. We shall take this cube as our unit; it contains eight carbon atoms. Since carbon is quadrivalent, it must contain 32 electrons ; these electrons will be situated a, at the middle points of the edges of the cubical unit: this accounts for 3; b, at the centres of each of the faces of the 8 small cubes: this accounts for 24; c, at the centres of the four small cubes not occupied by the carbon atoms: this accounts for 4 ; d, one at the centre of the large cube. Making use of this unit, we can calculate the electrostatic potential energy due to ‘the charges on the atoms and electrons. Jet E be the charge on a carbon atom, e that on an electron. The electrostatic potential energy of a carbon atom LE (= = r I find to be equal to : 1B ee Sica gee 9 7 (149 346.e—35'13.5), where 2d is the side of a unit cube. Since E=4e, this reduces to e2 IO) 71° The electrostatic potential energy of an electron I find to be se “42 119°340—147-59¢ | ‘ e? Seal 2d p: the Electron Theory of Solids. 661 Hence the electrostatic potential energy for the atom and 2 its four associated electrons is 21°15 _ Since there are eight atoms in the cube whose edge is 2d, if A is the density of the diamond and M the mass of a carbon atom, af A 1/3 or a = (a) ° Thus the electrostatic potential energy per one atom and four electrons is Av 13 Q115e (a) and the energy per unit volume is 4/3 21-15 (5) Hence, by page 736 of the former paper, & the bulk modulus of the diamond is given by the equation ee ED: SA oe — 9 é(5) : for the diamond A=3°52,. M=12x1'64x107-*4; hence jp ole Wir Ais x 10-2. This value for 1// is much less than that, °5 x 10~*, found by Richards. It is, however, in close agreement with "16 x 10~”, the value recently found by Adams (Washington Acad. Sc. xi. p. 45, 1921). The properties of solids formed by elements whose atoms have more than four disposable electrons are quite different from those of solids formed by the elements with one, two, or three disposable electrons. The latter are, with the exception of boron, metallic and good conductors of electricity and heat. The former, for instance sulphur and phosphorus, are insulators. Not only do they insulate in the solid state, but they do so after they are fused. They differ in this respect from solid salts which, though they may insulate when in the solid state, generally conduct when melted. This suggests that in the salts there are positively and negatively electrified systems which are fixed when the substance is in the solid state, but can move about when it is liquefied. In such elements as sulphur or phosphorus there does not seem to be any evidence of the existence of 662 Sir J. J. Thomson: Further Studies on anything but neutral systems ; in other words, the solid may be regarded as built up of units, each of which contains as much positive as negative electricity. It is noteworthy that according to the Hlectron Theory of Chemical Combination, two similar atoms if they have each more than four dis- posable electrons, like the atoms of sulphur and phosphorus, can combine and form a saturated molecule, which is electrically neutral. ? Thus we are led to distinguish three types of solids :— a. A type where the atoms are arranged in lattices, and the electrons in other lattices coordinated with the atomic ones. In this type each electron has no closer connexion with a particular atom than it has with several others. Thus, for example, when the electrons form a simple cubical lattice with the atoms at the centres of the cubes, each electron has 8 atoms as equally near neighbours ; so that an electron is not bound to a particular atom. This type includes the metals ; it also includes boron and carbon in the form of diamond, which are insulators. b. A type represented by the salts ; here the atoms are again arranged in lattices, but each electron has much closer relation with one particular atom than it has with any other. Thus to take the case of Na. Ol, where the Braggs have shown the atoms to be arranged according to the following scheme :— Na Cl Na C] Cl Na Cl Na Na Cl Na Cl We suppose that each sodium atom has lost an electron, while each chlorine atom has gained one; thus each chlorine atom has eight electrons around it, and each electron is much more closely bound to one particular chlorine atom than to any other. It is so closely associated that it is not dissociated from it; partner in either the solid or liquid state of the substance. Thus the chlorine system always has a negative charge, the sodium one a positive. These atoms do not move when the substance is in a solid state, though they may do so when it is liquefied. If the distance of the electrons from the chlorine atoms were to increase until it was not far from half the distance between the sodium and chlorine nuclei this type would approximate to type a. the Electron Theory of Solids. 663 c. A type where the lattices are built up of units which are not electrified ; such units are probably molecules containing two or more atoms, though in certain cases they may be single atoms. The characteristic of the type is that each unit has sufficient electrons bound to it to make it electrically neutral, and that each electron remains attached to a particular atom. Thus where an electric force acts on the system there is no tendency to make the unit move in one direction rather than the opposite, so that the substance cannot conduct electricity. Metallic Conduction. We now pass on to consider why it is that the arrange- ment of atoms and electrons in type @ is in many cases, though not in all, connected with the property of metallic conduction. ‘The consideration of the frequencies of the vibrations of the electrons in a lattice will, I think, throw light on this connexion. I showed (Phil. Mag. April 1922, p. 721) that these frequencies may extend over a very wide range of values as the type of displacement of the electrons is altered. Thus, if ali the electrons in a region whose linear dimensions are large compared with 2d, the distance between two electrons, have the same displacement relatively to the atoms, the frequency n of the vibrations for the alkali metals is given by the equation eC SC. ed aes OL) This frequency, even in the case of the univalent element, corresponds to that of light in the visible part of the spectrum ; for elements of greater valency it is far in the ultra-violet. This is also the frequency with which a single electron vibrates if the surrounding atoms and electrons are fixed. As those frequencies are so great very little energy will go into them at ordinary temperatures, and they will have little or no effect on the specific heat of the solid. There are, however, other types of vibration for which the periods may be very long. Thus if all the electrons ona certain line of the lattice are displaced along the lattice by the same amount, while those on adjacent lattices are displaced in the opposite direction, the frequency is given by the equation a eOt eee Bene" (2) mp Sr, ie = ga eae oc bt aie thus we see that only under certain conditions is the ex- pression for p? positive, and it is only under these that the OO 664 Sir J. J. Thomson: Further Studies on equilibrium is stable for this mode of displacement. A negative term will occur in the expression for p? if the electrons along one line of the lattice are displaced relatively to those on adjace:.t lines, even though the displacements are not equal and opposite. The view I wish to put forward is that in metals the frequency of this type of vibration is so low that the equilibrium for such a displacement is practically neutral, and therefore that a system vibrating in this way can absorb at any temperature the full amount of energy which at that temperature corresponds to each degree of freedom. It may be desirable to illustrate the argument by a particular case (fig. 1). Let us take that where the electrons Fig. 1. A -A A A are arranged in a simple cubical space lattice with the atoms at the centres of the cubes. Then, if an electron were dis- placed independently of the others, the frequency of its vibration would be very great and it would absorb very little energy; while if a chain of electrons along a lattice like AA’ A'’A'’ were displaced along the line of the lattice, the time of vibration of the chain might be comparatively ~ infinite, so that the chain would absorb the full amount of energy corresponding to one degree of freedom. However many electrons there may be in the chain, it has only one degree of freedom, for the nature of the displacement supposes that they move as a rigid body along a definite line. To sum up, the study of the frequencies of vibrations of the lattice of electrons shows that while at ordinary temperatures little energy could go into vibrations corre- sponding to the motion of an electron as a separate individual, yet groups of electrons along a lattice forming a rigid chain and moving in the direction of the length of the chain might absorb a full quantum of energy. Thus in a solid with the constitution we have sketched, chains of electrons lying along a line of a lattice may be travelling along that line carrying energy and electricity from one part of the solid to another; the frequency of the Electron Theory of Solids. 665 vibrations of these chains is so low that they readily absorb energy even at low temperatures, so that the average energy of the chains at the absolute temperature is KO, where k represents the factor corresponding to one degree of freedom. Thus, though the electrons in the solid are not free, and are in a very different condition from those of an electron gas diffused through the solid, yet like those in the gas thev can carry energy and electricity from one place to another. In the gas, howev er, each electron is supposed to be moving independently of its neighbour, and also to possess energy 3k@ corresponding to three degrees of freedom; in our case the agents which carry heat and electricity are not isolated electrons, but chains of electrons moving as if the electrons which compose them were rigidly connected together ; thus, however many electrons there may be in the chain, the average energy of a chain will only be £0, i. e. one-third of that of each electron on the gas theory. Thus on this view the contributions of the electrons to the specific heat of the solid will be a very small fraction of the contribution of the same number of electrons on the gas theory. Professor Lindemann has given (Phil. Mag. xxix. p. 127, 1915) a theory of Metallic Conduction which, though on quite different lines to the present one, agrees with it in making the electrons which carry the current move along the lines of the lattices, and in the view that the electrons make no appreciable contribution to the specific heat. The existence of these chains requires that the frequency of this vibration should be exceedingly small ; if the dimen- sions and arrangements of the lattice are such that the frequencies given by equation (2), are not less than 10” or so, the chains will not absorb energy at moderate tem- peratures, and at these temperatures the solid will act as an insulator. Thus it requires special conditions for the lattices of electrons to give rise to conductivity, so that the fact that neither boron nor the diamond is a conductor is not inconsistent with the theory. The motion of the chains need not necessarily be a reciprocating motion, for if the amplitude of excursion of an electron in the chain exceeds half the distance between two electrons, an électron such as A’ would shoot past another position of equilibrium; the forces acting on it would change sign and would tend to increase the distance still further; thus the chain would continue to move on in one direction and would not oscillate backwards and forwards. 666 Sir J. J. Thomson: Further Studies on On the Origin of the Chains. if we consider the state of things inside a solid, we can I think see reasons for believing that the existence of moving chains of electrons is probable. The solid is traversed by the radiation corresponding to the radiation from a black body at the temperature of the solid. This radiation consists of a series of discrete pulses, each pulse being the seat of intense electrical forces. When the effect of these pulses is represented by a Fourier series of waves, the wave-length for which the intensity of the light is a maximum is inversely proportional to the absolute temperature, and at 0° ©. is about 107? cm. We may suppose that the linear dimensions of the regions occupied by individual pulses are grouped about a mean which varies inversely as the absolute temperature, and which is large compared with the distance between two electrons. Thus the radiation will furnish fields of electric | force which have a high and fairly constant value over a length which includes a good many electrons; and the electrons in a lattice will from time to time be exposed to electric forces extending over a considerable length, and thus a chain of electrons will be started in motion as a whole. We should expect the average length of the chain to be inversely proportional to the absolute temperature. Moreover, such chains of electrons moving past the atoms would themselves tend to set up pulses of radiation, the dimensions of the pulse being commensurate with the length of the chain. Thus there would bea kind of regenerative action ; the radiation would tend to produce the chains, while the chains would tend to produce the radiation. When the two processes got into equilibrium the radiation would be that corresponding to the black body radiation at the temperature of the solid, while the average kinetic energy of the chains would be proportionai to the absolute temper- ature. When the solid is not acted upon by electric forces, there _ will be as many of these chains moving in any one direction as in the two opposite, so that there will be no current of electricity through the solid as a whole. The motion of the chains will give rise to “local currents” whose distribution might be affected by magnetic forces. . We shall now consider the effect of an electrics force on the motion of the chains. On the old theory that the electrons moved freely through the metal and kept striking against its atoms, the result of an electric force X was to give to the electrons an average the Electron Theory of Solids. 667 velocity in the direction of the electric force equal to XeA/2mv, where A is the mean free path of an electron and v the mean velocity. This result is obtained as follows: in a collision between an electron and an atom, since the mass of the electron is infinitesimal in comparison with that of the atom, there will be no “persistence” of the velocity of the electron. The velocity communicated by the electric force to the electron before it came into collision with an atom will, as it were, be completely wiped out by the collision, and the electron will make an entirely fresh start. Thus if ¢ be the interval between two collisions, the average velocity of the electron in the direction of the electric force will be tT iXe DER eX Bones 2m v On the theory we are now discussing, the carriers of electricity are not free electrons, but chains of electrons rigidly connected moving along a line of the lattice; since the chain has only one degree of freedom, the average energy of a chain at the temperature @ is R@/2 ; hence ees ee ss (8) where n is the number of electrons in the chain and v its velocity. Thus the average energy of a single electron in the chain is R@/2n. On the old theory when each electron was supposed to be free, its average energy was 36/2. The energy and velocity of an electron on the new theory are smaller than on the old. The ‘collisions’? between the electrons and atoms are also different. On the new theory an electron in a chain is moving past a row of atoms arranged at equal intervals 2c along a line parallel to the path of an electron; the time it takes for an electron to pass from closest proximity to one atom to closest proximity to the next is 2c/v. If the inter- change of energy between the electron and the atom were limited to the time when the electron was closest to the atom, the electron for a time 2c/v would not be losing any energy, and so could, under the electric force, acquire a velocity equal to Xe.2c/mv. The loss of energy by the electrons will not, however, be confined to the positions of closest proximity, but will extend some way on either side. The result of this will be that in part of the interval 2c/v the electron will be losing velocity, so that the velocity it will acquire under the electric force will be less than Xe . 2¢/mv, and the average velocity will be less than half this value. 668 Sir J. J. Thomson: Further prune. on We shall suppose that the average velocity due to the electric force is OE Ge as rae . ° ° ° ° ° . (4) where g is a fraction. This by equation (3) is equal to Xencv eXlv JRE —79Re°? where / is the length of the chain. If g is the number of chains parallel to w per unit volume, the number crossing unit area in unit line is equal to Xely g oRA 2? and since each chain carries ne units of electricity, the current across unit area is Xe’nly — 7 ORG Hence a, the specific electrical conductivity, is given by the equation f elung _e’lufp ls Rea OR ais. Late o: 0Re. oatee (5) where p is the number of electrons per unit volume and / the fraction of them formed into chains. On the theory we are considering, these moving chains are responsible not only for the electrical conductivity of metals, but also for the production and absorption of the radiation which fills the space occupied by the metal. They may be regarded as in some ways analogous to Planck’s oscillators, the slowly moviny ones corresponding to oscillators with a long period of vibration, producing mainly the long-wave radiation while the chains with high velocities give out the radiation corresponding to the shorter wave-lengths. We see from equation (3) that, at the same temperature, the chains which have a high velocity contain a small number of electrons and are therefore short, the chains which have a small velocity contain a large number of electrons and are long. Thus the long chains produce the long wave-length radiation, the short chains the short waves. We should expect on this view that the lengths of the various chains in a metal should be distributed according to a law analogous to that which governs the distribution of the energy corre- sponding to waves of different wave-lengths in the radiation from a black body. But according to Wien’s Displacement Law, the length-scale of the radiation varies inversely as the absolute temperature; 1,0=aconstant. Hence we conclude the Electron Theory of Solids. 669 that the average number of electrons in a chain varies inversely as the absolute temperature. If cn=£/@, where @ is a constant, then by equation (3) v= (3°) 0. Bm Thus nev is independent of the temperature except for the variation in c, due to the alteration in the volume of the metal caused by a change of temperature. jp, the number of electrons per unit volume, will only change with the temper- ature through thermal expansion. Hence we see from the ex pression (5) for the conductivity that if the number of electrons concerned in carrying the current does not vary with the temperature, the specific conductivity will vary inversely as the absolute temperature, which is sey approxi- mately true for pure metals. Resistance under rapidly Alternating F\ orces. We can get an estimate of the average velocity of the chains in the following way :—In the preceding investigation we have supposed that the electric force acting on the metal was steady. The argument will evidently not hold when the force is alternating so rapidly that while the electron is passing through the distance 2gc the force changes its direction ; tor in that case the effect of the electric field in altering the motion of the chains will be much less than that expressed by equation (4). When the force is reversed many times during this period there will be very little alteration, and therefore very little conductivity. Thus the resistance of metals under alternating forces should begin to. increase when the period of alternation becomes comparable with the time taken by an electron in a chain to travel over a distance equal to g times that between two electrons in the chain. When the period of alternation is considerably greater than this time we should not expect the resistance to. vary with the period. Rubens and Hagen determined the conductivity of metals under alternating “forces by measuring the amount of light of very long wave-length reflected from the surface of the metals. They found that the electrical conductivity of certain metals at room temperature under electrical waves. whose wave-length was 2°5x 107° cm. was the same as the conductivity under steady electrical forces, and that even when the wave-length was as short as 4x107* cm. the electrical conductivity was within about 20 per cent. of that. 670 Sir J. J. Thomson: Further Studies on for steady forces. As the period of the longer waves is 8:3 x 10714 second, we may conclude that the time taken for an electron in a chain to pass over g times the distance which separates it from its next neighbour in the chain cannot be greater than about 107 second. If we take the distance between 2 electrons as 2xX107%8, this would make the minimum velocity of the chains about 29x 10°. This refers to the temperature at which Rubens and Hagen made their - experiments—presumably about 15° C. As the velocity of the chains decreases as the temperature falls, the reflexion from a metallic surface should become at very low -temper- atures abnormal at longer wave-lengths than those determined by Rubens and Hagen. We can get in another way an estimate of the magnitude of the time taken by an electron in a chain to pass over a distance equal to half the distance between two neighbouring electrons in the chain. At the temperature of 15° C. the wave-length of the light of maximum intensity in the black body radiation is 107? cm.; the time of vibration of this light is }x 107% see. We should expect from the way we have supposed the black body radiation to arise, that this time would be of the same order as that taken on the average by an electron in the chain to pass over g times the distance between two electrons, and so again we arrive at 107 sec., as being a time of this order. On the supposition that c/v is proportional to the time of vibration of the light of greatest intensity, we have eens Vins where y is a constant which does not depend on the metal. nv Hence N= ae but imnv? =4R8, m being the mass of an electron ; thus nv’= #9= 15x 10" x. 7 Thus Ei —leodusicy x Ores The specific conductivity e*ncv = ofp RO | sO) oe WO hy oipee. es RO ; the Electron Theory of Solids. 671 in this expression the only factor which varies from one metal to another is fp, the number of electrons made up into chains ; the conductivities of metals at the same temperature are directly proportional to the number of electrons in unit volume which take part in carrying the current. We can put the expression for the conductivity in the form ec a mv If we take ge/v at 15° C. to be 107", since é?/m=2°8 x 10738, the conductivity at this temperature is equal to hp eo SLO The values of f calculated from this expression for some metals are given in Table I. TABLE I, Metal. sap ap Dp. f. Babitunn, h,, dads: 1:1x107* 25 x 10? ‘16 Sodium ...... ce ee! aoa Mont tg ‘47 Potassium ..... Rees tye SO 7s ‘70 Bybidium. ..... 03 ..08 “See *O6:..,, 7510) Ch ante 048 ,, 85 olen net. 9283 ES. 25 19 Magnesium ......... DSi ty aS “1% Shin ee ae CER IGDs: & ho. ,: 08 Oadmmme-......-... PS: ee 33) ) ee ‘09 POMP <...3.... 3 ds Tas HE Oe =. a 4 Thus on this theory, potassium has a much larger per- centage of its electrons moving about in chains than any other metal. To form an estimate of the average number of electrons in a chain and the velocities of the chains, we may proceed as follows :—If we suppose that at 15° C., gc/v=107, then if c=10~*, which is about right for sodium, v=10° xg. When v is known, we can get n from the equation tmnv?=tRO. If v=109, this equation gives n=4°4 x 10/9’, As g must be less than unity, the chains at this temperature 672 Sir J. J. Thomson: Further Studies on will on the average probably contain more than 10,000 electrons, and their average length would be greater than 2.10-4em. As the average length of the chains varies inversely as the temperature, the average length at 3° Ab. would be greater than 02 cm. and their average velocity less than 10° cm./sec. Super-Conductivity. The expression for the specific conductivity given by equation (5) is based on the assumption that in a “ collision ” between an electron and an atom, the energy imparted to the electron by the electric field is given up to the atom during this “ collision,” so that the electron starts as it were afresh after each collision. For this to happen there must during the collision be a considerable transference of energy from the electron to the atom. The energy of the atoms is due to their vibrations about positions of equilibrium, and the frequencies of these vibrations, according to the experi- ments of Nernst and Lindemann on the variation of the specific heats with temperature, range from 10” to 10% for the different metals. Now, it follows from general dynamical principles that a collision lasting for a time which is long compared with the time of vibration of a system, will excite very little vibration in the system and communicate very little energy to it. The amount of energy communicated | will fall off very rapidly as the ratio of the duration of the collision to the time of vibration increases Ina case con- sidered by Jeans, ‘ Kinetic Theory of Gases,’ § 481, the energy communicated to the system was proportional to e *?, where c is the duration of the collision and p the fre= quency of the free vibration of the system. It follows from this that when the chains of electrons are moving so slowly that the time of a collision is long compared with the time of vibration of the atom, very little energy will be trans- ferred. Our expression for the electrical conductivity was, however, obtained on the assumption that at each collision the excess energy due to the electric field was given up. If, however, the transference of energy is not sufficient to allow of this, the average velocity of the electrons will be greater than that calculated, and the conductivity greater to a corresponding extent. If there were no transference of energy, the average velocity of the electrons and the electrical conductivity would both be infinite. We see then that when the temperature gets so low that the time taken by an the Electron Theory of Solids. 673 electron to pass over a distance 2¢ is comparable with the time of vibration of the atom, any diminution in the temper- ature will produce an abnormally large increase in the conductivity, and thus the metal would show the super-con- ductivity discovered by Kammerlingh-Onnes. The numbers we have just obtained for sodium show that at a temperature of 3° Ab. the time taken by a collision would be greater than 2x 107", and this is very long com- pared with the time of vibration of the sodium atoms, which have a frequency of 3°96 x10". There would be very little transference of energy at this or even considerably higher temperatures, so that the conductivity would be very oreat. ” We have associated the time taken by a chain to pass over the distance 2c at any temperature with the time of vibra-_ tion of the light of predominant energy at that temperature. On the theories of the variation of specific heat with temper- atures given by Nernst, Einstein, and Debye, this variation isa function of the ratio of the time of vibration of this light to the time of vibration of the atom. Thus on the view that the average time of a collision is about that of the time of vibration of this light, the variation of the specific heat with temperature and the communication of energy from the electron to the atom depend upon exactly the same quantity, and thus the variation of the specific heat with temperature ought to be closely connected with the super-electrical conductivity. The product 0c of the temperature and the electrical conductivity ought to change rapidly with the temperature when the specific heat does so. The product 0c will increase as the specific heat diminishes ; if, however, we were to plot the reciprocal of 0c against the tempe- rature, we should expect to get a graph very similar to the one representing the connexion between specific heat and temperature. That a connexion of this kind does exist between 1/0c and the specific heat is, I think, shown by Table II., which ‘contains the values of 1/@o for lead and silver calculated from the values of the resistances given by Kammerlingh- Onnes (communications from the Physical Laboratory of Leiden, cxix. 1911); the third column contains the values of 6/0 when @=hy/R, where N is the time of vibration of the atom ; the fourth column gives the value of the specific heat calculated by Debye’s theory (Jeans, ‘ Kinetic Theory of Gases,’ §§ 553) ; and the fifth column the ratio of 1/00 to the specific heat. Phil. Mag. 8. 6. Vol. 44. No. 262. Oct, 1922. 2X 674 Sir J. J. Thomson: Further Studies on TaBLeE II. Lead, @=95. Specific Ratio of 1/0¢ 6, 1/00. 6/0. heat. to specific heat. 273 366 2:88 993 868 169°3 351 1°78 “984 357 779 325 +82 928 350 20°18 150 ‘215 ‘41 365 13°88 87 145 2 435 Silver, @=215. 278 366 1:27 ‘965 380 169°3 3438 ‘79 924 Be 77-9 252 362 ‘691 365 20:18 45 ‘095 ‘073 615 Thus_except at the lowest temperatures the ratio of 1/A¢ to the specific heat is fairly constant ; and inasmuch as Kammerlingh-Onnes and Clay have shown that when a small amount of impurity is present, the resistance at very low temperatures approaches a finite value instead of continually — diminishing as the temperature falls, it is evident that at these temperatures a trace of impurity would produce a large increase in the value of 1/@c. The higher the value of ©, the higher will be the temperature at which an abnormally large increase of the conductivity with fall of temperature sets in. Of all metals, beryllium has the smallest atomic value, and so we should expect it to have the greatest value of vy and ©; it seems probable that the temperature coefficient of this metal may be abnormal even at room temperatures. Thermal Conductivity. The motion of the chains of electrons along the lines of the lattices will in an unequally heated conductor tend to equalize the temperature, for much the same reason as on the Kinetic Theory of Gases the conduction of heat is brought about by the motion of the molecules of a gas. There are, however, several points of difference which require . discussion before we can proceed to find an expression for the thermal conductivity on the chain electron theory. When the temperature is uniform, there is no ambiguity in the statement that the average kinetic energy of the chain is that corresponding to one degree of freedom. A chain of electrons, however, stretches over a distance large compared with the distance between two atoms, and when the temper- ature is not uniform the temperature at one end of the chain may not be the same as that at the other, As the the Electron Theory of’ Solids. 675 electrons in the chain move like a rigid body, each electron has the same kinetic energy; we shall suppose that this energy is the same as if the whole of the chain were at the temperature of its middle point, so that the kinetic energy of the whole chain is that corresponding to one degree of freedom at the temperature of the middle point of the chain. Another important point is that the energy carried across a plane by a chain of electrons passing right across it may, when the temperature is not uniform, be much greater than the actual kinetic energy in the chain when it first reaches the plane. This is important because if it were not so the transport of energy due to the motion of the chains would not be great enough to account for the observed thermal conductivity even it every disposable electron were utilized to make up the chain. It must be remembered that on this theory the number of disposable electrons in unit volume is known; for example, in the alkali metals it is equal to the number of atoms, and cannot be regarded as a quantity whiclr can be adjusted so as to give the right value to the thermal conductivi'y. To see how this additional transport of energy is brought about, consider what happens when a chain “of electrons ABCDE crosses the plane ZZ, moving past the atoms in its neighbourhood and exchanging energy with them. ~ If 2¢ be the distance between neighbouring electrons or atoms, we shall define a collision between an atom and an electron to be the passage of an electron past its shortest distance from the atom. If we take the axis of x parallel to the chain, then when the head A of the chain reaches 7Z each of the electrons in the chain has 1/n of the energy corresponding to one degree of freedom at-the temperature l dé tid 2 dx’ its electrons, and @ the temperature of the plane ZZ. When A makes a collision with the atoms just to the left of the plane ZZ, it will momentarily lose an amount of energy 1/dé proportional to = qa This will lower its energy below that n which must be possessed by every electron in a chain whose middle point is now at a place where the temperature is where / is the length of the chain, n the number of 6+4(1—2c) = this energy only differs from that before . 2c dd ' impact by onda? the electron has, however, since / is much uv ae 2 676 Sir J. J. Thomson: Further Studies on greater than 2c, lost far more than this, so that energy must be transmitted along the chain to A to bring its energy up to its proper value. Thus when the electrons are connected together in chains, the transference of energy is not confined to the energy carried by the electrons when they are crossing the plane ; each collision made by an electron in the chain will, until the whole of the chain has passed the plane, result in the transference of energy across the plane; if the chain is long this second type of transference may far exceed in magnitude that which would occur if there were no collisions. We shall now proceed to find an estimate of the trans- ference due to the collisions. Let us take the electrons in the chain in pairs, the con- stituents of a pair being equally distant from the centre; let this distance be y. Then, as the chain moves along, one of the constituents of the pair has energy corresponding to a temperature 7 = e dé corresponding to the temperature Use below the temperature of its position. If $R@ is the energy corresponding to one degree of freedom at the temperature 0, this excess or defect nee We shall suppose that at each collision of an electron with an atom the energy of the electron is restored to the value corresponding to its position. | Let us begin with the electrons at the beginning and end of the chain. We have seen that the first collision of the front electron after passing the plane results in the trans- above, the other constituent the energy of energy of an electron will be l ference of a units of energy across the plane. The collision made by the electron in the rear will result in its gaining Tors units of energy ; this will have to be given up by the chain, but inasmuch as all the chain is on the right of the plane ZZ, the energy will be given off in this region and will not be transferred across the plane. Thus the first collision of this pair of electrons transfers R1 dé An dx units of energy across the plane. Let us now consider the next collision. The front electron the Electron Theory of Solids. 677 will lose - os units of energy, and this will have to be n dx 5 supplied from the chain ; since part of the chain is now on the left-hand side of the plane, some of this energy will come from this part, and will not be transferred across the plane. The energy coming from the part of the chain to the right will be transferred across the plane ; the ratio of the length of chain to the right of the plane to the length of the chain is (l1—2c)/l. We suppose that this fraction of the whole energy comes from the part of the chain to the right, and so is transferred across the plane. ‘Thus the transference of energy due to the second collision of the front electron is Rl dé [—Ie 4ndx ° Now consider the second eollision of the electron at the rear of the chain. This electron will by the collision receive a = units of energy, and as there are a large number of electrons in the chain practically the whole of this must be given out again by the chain. If it is given out uniformly from all parts of the chain, since the length of the portion to the left of the plane ZZ is 2c, the amount of energy given out in this region, which is the amount transferred across the plane ZZ, is IR dé 2c Anda’ I Thus at the second collision of this pair of electrons the energy transferred across the plane is equal to IRd@ 1—2c IR dO 2% Lee Lede. | IR dé ~ An dx’ the same as that transferred at the first collision. We can see that this must be true of all the collisions; and as there are n of these before the chain gets right across the plane, the total amount of energy transferred across the plane by the collision of this pair of electrons is equal to IR dé 4 dx For a pair of electrons at a distance y from the centre of the chain, the interchange of energy at each collision with an atom is 2 = and the number of collisions with one Qn dx 678 Sir J. J. Thomson: Further Studies on member of the pair in front of the plane is equal to 2ny/l; alt hence the energy transferred by this pair is = ue Thus, giving y all possible values, we find that the total amount of energy transferred across the plane ZZ through the collisions | of all the electrons in the chain is dG rl. \7 l : l \7 My eG) +G-a) +) <= =. —,— when x Is large. 24. dea their average velocity, the energy transferred across unit area per second is | 7 7 nl dé 24 ai dx’ In making this rough estimate of the transference of energy, we have supposed that the transference occurred only when the electron was in closest proximity to the atom. The process by which the electron first loses energy to the atom and regains it again by a transference of energy along the chain will begin before the electron reaches its shortest distance from the atom and go on after it has passed it ; the result of this will be that at each passage of an electron past an atom the transference of energy may be very con- siderably greater than that in the case we have considered. We must therefore suppose that the transference of energy y Rdé | ! at each collision is not 5 —— but a multiple of this, viz. Al Ax ey Rd@ 2n dx’ where eis a number greater than unity which depends on the law of force between the electron and the atom. This will make the transference of energy across unit area per second equal to IL dé ae ee where ¢ is the number of chains per unit volume and » the velocity of a chain. Hence K, the thermal conductivity of the metal, is given by the equation 1 | ae i enlguR igre 54 efplvh, the Electron Theory of Solids. 679 The ratio of K to o, the electrical conductivity, is given by the equation oir Geyer? i beg aes The right-hand side of this equation does not involve any quantity peculiar to the metal; hence the ratio of the thermal to the electrical conductivity should at the same temperature be the same for all metals, and at different temperatures should be proportional to the absolute temperature. This is the well-known law of Wiedemann and Franz, which is obeyed with fair accuracy by many metals. Summary. This paper contains a calculation of the compressibility of a divalent element, calcium, and also that of the diamond by the method given in my paper on the Hlectron Theory of Solids (Phil. Mag. April 1921). The results obtained are in good agreement with those found by experiment. The same theory is then applied to the consideration of metallic conduction, electrical and thermal. It follows from the theory that when an individuai electron is displaced relatively to its neighbours, the frequency of the vibration is that cor- responding to the visible or ultra-violet part of the spectrum ; these vibrations would not, unless at extremely high temper- atures, absorb an appreciable amount of energy. When, however, instead of a single electron being displaced, a chain of electrons lying along one of the lines of the lattice is displaced as a rigid body relatively to the neighbouring atoms and electrons, the time of vibration of this chain may be very long, so long that even at very low temperatures the chain may acquire the full quantum of kinetic energy cor- responding to one degree of freedom at its temperature. Thus chains of electrons moving like rigid bodies may travel along the lines of the lattices, and carry electricity and energy from one part of the metal to another. The theory that electric and thermal conductivity is due to the movement of these chains is worked out, and is shown to account for the variation of electrical resistance with temperature, for the super-conductivity of metals at very low temperatures dis- covered by Kammerlingh-Onnes, and for Wiedemann and Franz’s law of the proportion between electrical and thermal conductivity. [= 680.4 LIX. The Decrease of Energy of « Particles on passing through Matter. By G. H. Henperson, Ph.D.* ~§ 1. Introduction. FXHE general laws governing the passage of @ particles through matter have been discussed theoretically by both Darwin + and Bohr f. If E, M, and V be the charge, mass, and velocity of the a particle and e and m be the charge and mags of an electron, then, when the « particle approaches an electron along a line at a distance p from it, the energy given to the electron is, by the ordinary laws of dynamics, 2H? : Q= mV? (pe a a?) 5 ° ° ° ° . (1) a2 He(M-+m) is: Ke where . = MmV2) > mye thie if the electron is free. In passing through a thickness Aw of matter, the number of encounters in which p lies between p and p+ dp is 2a NnAxp dp, where N=the number of atoms in 1 em.’ and n=the number of electrons in one atom. Then, if T is the energy of the « particle, a iy 4 H?e? Nn | pdp De NO Neth iae If the limits of pin this integral be taken as 0 and w, the | integral becomes infinite, 7. e. an a particle could not pass through an appreciable thickness of matter atall. Hvidently some upper limit to the radius of action of the & particle must be taken. In the first paper dealing with the motion of « particles, Darwin made the assumption that the effect of the @ particle at any instant was confined to the electrons of the atom through which it was passing. He was able to calculate the motion of the @ particles through matter for various arrange- ments of electrons within the atom. Theoretical velocity curves showing the yariation of velocity with distance * Communicated by Sir EK. Rutherford, F.R.S. + Darwin, Phil. Mag. xxiii. p. 901 (1912). t Bohr, Phil. Mag. xxv. p. 10 (1913), and xxx. p. 581 (1915). (2) Energy Decrease of « Particles passing through Matter. 681 travelled were obtained which showed the same general form as the experimental curves, and from these an estimate of the number of electrons in the atom was made. On the other hand, Bohr considered the time of passage of the 2 particles past an electron to be the determining factor, and assumed that as long as this time of passage was small, compared with the period characteristic of the electron in dispersion phenomena, the electron could be considered as free. When the time of passage was comparable with this, however, the electron could no longer be considered as free, aid in this way an upper limit to p was introduced. The calculated velocity curves showed good agreement with experiment over most of the range of the 2 particle. § 2. Method adopted. In this paper the law of decrease of energy of an a particle is developed along lines somewhat different from those of the writers mentioned. | According to the modern ideas of atomic structure, due to Bohr, the electrons are thought to be arranged in various stationary states or energy levels. An electron leaving one of these stationary states can only move to another such state or completely out of the atom (to infinity). Thus the energy which an electron can take up is limited toa number of finite amounts characteristic of the atom. These views furnish a simple method of fixing the upper limit to the radius of action of an @ particle upon the electrons of matter. If the electron is to be moved from one stationary state to another by the passage of an « particle near it, then with the finite amount of energy which the electron must take up in order to effect the change there may be associated an upper limit to the radius of an & particle upon the electron. The mechanism which is involved in this transfer of energy from « particle to electron may be difficult to conceive. However, when an electron is moved from its stationary state to infinity by the action of light, the frequency v of the light must be such that the quantum hy is greater | than the finite difference of energy between the initial and the final states of the electron. Here the rule governing the transfer of energy is known, although the mechanism involved is not. A similar statement holds for the case when the transfer of electrons from one state to another is caused by electron impacts (elastic and inelastic impacts). Similarly, in the case of « particle impacts the transfer of 682 Dr. G. H. Henderson on the Decrease of energy a) be determined by appeal to experiment before the mechanism is understood. Accordingly, it seemed of interest to apply the classical theory of the exchange of energy as given by (1) to the case of the « particle and electron, having regard to the limited number of stationary states which the electron can occupy within the atom, and to compare the resulting law of motion of the « particle with experiment. This has been done in the following paper, taking asa basis the following assumptions. Interchange of energy with an electron takes place according to (1) provided that the energy transferable, according to (1), is greater than the ionization potential of that electron. hus for any given V a definite upper limit is placed upon p by (1), where Q is equal to the ionization potential. For values of p less than this limiting value po, the excess of energy over that required to remove the electron from the atom may be in the form of kinetic energy of the electron. The existence of resonance potentials is taken into account by assuming that when the energy available according to (1) lies between the ionization and resonance potentials, or between two resonance potentials, the energy transferred is constant and equal to the lower resonance potential. For encounters where p is greater than po, given by (1) for the lowest resonance potential, it is assumed that practi- cally no energy is transferred to the electron, the latter con- tinuing to move in its stable orbit and behaving as if rigidly bound to the atom. § 3. Calculation of the Law of Motion of « Particles. Consider a substance in each of the atoms of which there AVe Ny, Ny... ny electrons with the ionization potentials Q,, Qe . Q, respectively. The total number of electrons N=Ny+Ng+...0Nn Then, for the n,; electrons having the ionization potential 1» P| AT _ 2An 4 pdp Av =V? pita’ = > ) ‘ ‘ : 4 206 Energy of « Particles on passing through Matter. 683 where p; is given by 2 Hie? 9 9 +o / mV" 2), and ' paar Hence —-= —,- log y PY Aw VA a An, 2mV? | ——— 10D 72 ‘Saison AO ag Summing for all types of electrons, AN Gia ie ImnV? Rey ays Gk = Ey log 2mV2—X,n; log Q: | is the rate of loss of energy due to ionization potentials. To take into account the effect of resonance potentials, consider, first, the n,; electrons with the ionization potential. Q,. Let there be resonance potentials Q,', Q,’..., all less than Q,, and let the corresponding upper limits, given by (1), for the p’s be p,', p;''.... We assume that for values of p lying between p; and pj’ (t.e., when the energy available, according to (1), lies between the ionization and resonance potentials) ite energy transferred to the electron is constant and equal to Q,’. Similarly, for all values of p between Pr and p,'', the energy transterred is constant and equal to Qu" : Then the total loss of energy by the « particle passing through a distance Ax, which is due to the presence of resonance potentials, will be pt Pes AT=27Nn, Ax [a fray + Qu" {ody + a | ; : Py P,’ ae aNn [Q K 12 aD) Q a TED S ay HY ae | aE 11 Qi (p1°— pi") + 1 (Pi a a i 2a AT 27NE’e’n, ae A Pe 1 ) | Bet mv L&(Go-g) +"(Q7- gt) An ; Cer) =ViFq1- Qo f 684 Dr. G. H. Henderson on the Decrease of Summing this expression for all the types of electrons Ny, Ng, etc., we have ey oe = V3, a 3e4 1 = ar} Thus the e, expression for the loss of energy is (é+1) a == 7 [ leg V? 4 log2m—3.— oa _ log Q:+ 2s 3,41— ars | (+1) Put log b= log 2m —¥, “log Q,+> oS te 12 (3) Then Ap = v2 dV An fae the negative sign entering because AT is a loss of energy. MV2dV Thus dz= — An log V3 ION OND) 2Anl? log bV? M e-Ydy Me Si. ao SME PME ae where ha — 2 log dV’. Let the velocity of the a particle initially be V) and the velocity of the « particle after doing a distance # be V, then es Mee Mee DAsib? \ omy eee Yo where Y = -—2 log 6V?= — log b?V4, Y= — log b?V,'. M Hence C= 9Anb? [ Ee — Yo) — Hu(— Y) {| Sie (4) Hi(x) is the exponential integral, defined by aby Real {= . @ numerical values of which have been tabulated by various writers. Energy of « Particles on passing through Matter. 685 It is very interesting to observe that this equation is of the same type as that derived by Bohr in his second paper on the motion of a particles through matter, although derived on quite different assumptions. ‘The meaning of some of the constants is, of course, quite different. §4. Comparison with Hxperiment. Substituting accepted values of the physical constants we have from (4) 3 =e = Vay bat ¥) | for air at 15°C. and 760 mm. pressure, assuming the number of electrons in the fictitious air molecule to be 14:4. We substitute numerical values in expression (4) for log 6, term by term, log 2m= 2°30 logy) 2 x 9°0 x 10-2 = — 61°58. The remaining two terms are more difficult to evaluate, as the values of the ionization and resonance potentials are not completely known, and we are treating with average values for air. The order of magnitude of these quantities is, however, fairly well established. We will choose values of this order of magnitude which give the best agreement with experiment. Assuming 4 electrons for which Q=200 volts=3°18x 10-” erg and 10-4 electrons for which Q=15 volts aero LQ 10: ergy, Be tis 10°4 ii —3,"" log Q.=—23 | rq login 2°38 x 10- + 7qqlog 3-18 x 10- we) = 28°78. For values of the resonance potential which are near the ionization potential the terms (1—Q&*)/Q(”) will be practi- cally zero. Hence we need only concern ourselves with those few resonance potentials which are considerably lower than the ionization potential. We shall probably not be far wrong if we set X, (1—Q¢+)/Q=2 for each set of electrons. Then log b= —61°58 + 23°75 + 2:0 = — 35°83, log b?= —71°66, corresponding to a value of b?=7°5 x 10-* approximately. Thus the velocity equation of the « particle becomes #=7°79 x 10-5 [Hi( — Y,) — Ei(—Y)]. 686 Dr. G. H. Henderson on the Decrease of It should be pointed out that since } appears as well in the exponential integrals, the value of wis not very sensitive to changes in 0. Values of the range given by the formula have been calculated for various velocities of the @ particle, and the results are shown in the second column of Table I. The evaluation of K7(—Y) has been carried out By interpolation from the tables given by Jahnke and Emde*. In column three are given the experimental results of Mavsden and Taylor +. Column four shows the ranges as calculated by Bohrt. The Table refers to RaC in air vat 15° 0, MW ocimogs & V/V. Cale. Experimental. Cale. Bohr. 20 1°87 1:90 i ako) "8 3°43 aoe OO =i 4°50 AA 4°45 (5) Dea Doe oo 0 5°67 Daag 57 From the table it will be seen that the calculated values agree well with experiment. Both series of calculations fail for low velocities of the @ particles. Bohr’s theory holds down to values of V/V, equal to about -5; the present theory has not quite the same range of applicability, failing below values of V/V of about ‘6. One or two points of interest should be noticed here. First, we shall see what is the actual size of the radius of action of the a particle called for.: Taking an a particle moving with the initial velocity of radium C (1°92 x 10° cm./sec.) and a resonance potential of 10 volts, when numerical values are substituted in (1), we obtain Po Pe BO C107 eg. oo Ome and hence Po = 1°83 x 107%, Thus, ~o= 2°80 x 10-° cm. is the distance from the electron within which an « particle must come in order to transfer to it energy corresponding to 10 volts. This distance is of the order of one-tenth of the diameter of an atom. Secondly, there wiil be a velocity below which the @ particle will be unable to ionize, however close the collision. Asthe velocity of the « particle decreases the value of yo increases. * Jahnke u. Emde, Funktionentafeln, p. 19. |} Marsden and Taylor, Proc. Roy. Soc. A. 88, p. 4438 (1913). + Bohr, Phil. Mag. xxx. p. 597 (1915). Energy of « Particles on passing through Matter. 687. at first in inverse proportion to the velocity. A time will come, however, when the term in a? will become important, the value of po will then begin to fall off rapidly to zero ata finite value cE V, which may be called the critical velocity V.. For the ratio (9 + a*) /a2=2mV?/Q must not be less than unity. When it 1s equal to 1, 7% ban Then V2=Q/2m=8'8 x 10%, The critical velocity V-=9°4 x 107 em /sec. The same results should hold for positively charged hydrogen atoms, which should cease io ionize a gas of 10 yolts ionization potential at velocities less than about LO*em./see., equivalent to about 5000 volts. ‘This point has been discus downwards these variations become marked. Neither has account been taken of the orbital velocities of the electrons which may become appreciable for low & particle velocities. Experiment shows that the -beam becomes anhomogeneous in velocity. Straggling becomes very large at this point. Further, recent results obtained by the writer give evidence to show that the charge on the a particle does not remain invariable for low velocities, though further discussion of this point must be reserved. All things considered it seems clear that the behaviour of a beam of a particles becomes much too complicated at low velocities to be dealt with by simple treatment. Summary. In this paper the equation of motion of an «@ particle passing through matter is developed, making use of the concepts of resonance and ionization potentials. The equation found is shown to give good agreement with experiment in the case of air, but does not furnish a decisive test when compared with other solutions which have been proposed. Cavendish Laboratory, July 1922. Bee LX. A Kinetie Theory of Adsorption. By D. C. Henry *. 6a to the present time no theory of adsorption has been developed which leads to equations valid over the whole range from low to high concentrations. The adsorption, both of gases and of solutes from solution, is well expressed for low concentrations by the empirical ‘ exponential formula ” 1 Timely a where « denotes the quantity adsorbed, ¢ the exterior con- 1 centration, and & and 5) are constants. As soon as moderately U high concentrations of adsorbate are reached, this formula gives results greatly in excess of the values observed, which appear to tend to an upper limit. In the present paper a theory of adsorption is developed based on the conceptions of surface action introduced by Hardy and Langmuir. An adsorption equilibrium is con- sidered as involving a balance between the rate at which molecules of adsorbate condense on the surface of the adsorbent and the rate at which molecules leave, or evaporate from the same surface. The fundamental assumptions made are two, for both of which Langmuir has produced much evidence. In the first place, it is assumed that the range of action of the forces which bind molecules of adsorbate on to the adsorbing surface is comparable with the diameter of an atom, so that the layer of adsorbate molecules bound by the field of force of the adsorbent will be only one molecule thick. Secondly, it is assumed that the impact of a molecule on a surface is completely inelastic, so that every impinging molecule will condense. General Adsorption Equation for n Gaseous components. Consider an adsorbing surface, of area w, brought into contact with a homogeneous gaseous phase containing components §j, Sg,...S,. Whether the surface be crystalline or liquid, it will present a more or less regular arrangement of points of unsaturated field of force, where molecules of adsorbate can condense; it the surface is crystalline, the arrangement will be related to the crystal lattice, if it is liquid, to the packing of the oriented surface molecules. * Communicated by Prof. S. Chapman, F.R.S. Phil. Mag. 8. 6. Vol. 44. No. 262. Oct..1922, 4 ¥ 690 Mr. D. C. Henry on a Suppose there are Ny, such points of attachment. If a molecule from the gaseous phase impinges on a point of attachment unoccupied by any other adsorbed molecule, it will condense, forming a single ‘adsorbed layer’; if it impinges on a point already occupied, it will also condense, forming a second layer. But the relative life of a molecule on the surface will depend on the attractive force exerted on it by the surface, and if, as is usually the case, the attractive force between molecules of adsorbent and adsorbate is much greater than that between two molecules of adsorbate, the relative life of a molecule in the second layer will be so small compared with that of a molecule in the first layer, that we may treat molecules impinging on points aiready occupied as if they were immediately reflected and never condensed. — At any moment let a fraction @ of the points of attach- ment be vacant, and fractions 6), 0,,... 9, be covered with monomolecular layers of 8;, 82, ... 8, respectively; then Oo O)4 0, +>... Fez 1c. oo eee Let a single molecule of 8), Ss,...8, occupy respectively a1, Ag, ... dm points of attachment *; the number of molecules of component 8S, adsorbed will be ea, and the adsorbed quantity in gram-molecules will be f : 0,No oD) LN? eo x= where N is Avogadro’s constant (6°06 x 1075). Now the rate of evaporation from the surface will be determined by Maxwell’s distribution law as the number of molecules which reach, per second, a state of agitation sufficient to break free from the force field, and is given, for §,, in gram-molecules per second, by the expression Tt Ay AGN Bee RE. X= (ay) v.Xaoe ee ee (3) where 2X, is the internal heat of evaporation of §, from the surface, A, is a constant depending on the field of force, and R and a have their usual meanings; vy, is written for the expression Agee ae) | BT * Since the number of points of attachment occupied by a molecule is not subject to a merely geometrical restriction, but is determined by the field of force, it does not appear essentia} that the quantities a, should be integers. ‘ + Langmuir, Jour. Amer. Chem. Soc. xxxy. p. 122(1913). Ri ] Phil. Trans. A. cei. p. 501 (1908), Pe ae 691 Now if p, is the partial pressure of §, in the gaseous phase, the rate of impact of molecules of this component is * Kinetic Theory of Adsorption. ] ———__ Xx p. om.-molecules per second per sq. cm. oo Pr p per sq.em,, where m, is the inolecular weight of §,; hence the rate of impact on the surface of area w is w times this quantity, which may be written (4) For a molecule of 8, to condense, ‘+ must impinge on a spot where there are a, vacant adjacent points of attachment. The chance that a given molecule will find one point vacant is 0), and the chance that it will find a, adjacent vacant points will be 0), if we neglect the possibility that the molecule may need to impinge in an orientation related to the configuration of the vacant points+. Hence the rate of eondensation of molecules of S, is @u,p, gm.-molecules per second. . (9) For equilibrium, the expression (4) must equal the expression (5), and we get the n equations @,p,9)” gm.-molecules per second. . ae vere 1 VX, = Opp prOy” or, from (1) | X,= op(1— (a ee ae a which from (2) Sa aes ae |. = pol N, Xy ane Ny Des xy Sin tie = 67rP- f= xa! ea Danis p) (6) where € is written for APO ae fir os Oy, apnea Len A) ae aR: yl) ome Se aera Od) N and = aN’ (8) and is therefore the saturation capacity of the surface for S,, supposing it to be completely covered by a monomolecular layer of that component. : * Jeans, ‘Dynamical Theory of Gases,’ 1916, p. 133. Phys. Rey. i. p. 831 (1918). + This possibility could probably be allowed for by multiplying Oy by a constant depending on @,, which could then be included in ¢... AX 2 Langmut, 692 Mr. D. C. Henry on a The n relations (6) determine the equilibrium adsorptions of the n components. In a similar manner we can write for the peau Ny of adsorption of S$, in gm.-molecules per second At, Ln rr = Wy Py dee X, aT eae EON xe a ° 1) where #, is the instantaneous value of the adsorption of 8, at time ¢, and p, is the instantaneous value of the corresponding partial pressure. A similar relation holds for each of the nm components. The Temperature Coefficient of the Isotherm. The effect of temperature on the equilibrium adsorption follows from equation (6), the only constant of which that involves the temperature being &, which from equation (7) is given by A fata -e8, where ¢ is a constant ee of the temperature. The qualitative conclusion that adsorption decreases with rise of temperature follows immediately. The relations (6) and (9) do not admit of general solutions. Solutions must therefore be obtained for special cases. One Component only— Adsorption of a single Gas. _ Equation (6) reduces to X=tp[1— ei) on ol which can be expressed x x In = Ing-+ a. In| 0 (12) NR od ewe =Inf—a| 5) +5 c te | For moderately small adsorptions we can use the approxi- mation | Xx i In - =Inf— xi xX cg ON ae | : or lee = log €—0°4343 . XT° D Pap cme 0) (115 The relation (13) is of the same form as that obtained by Kinetie Theory of Adsorption. 693 Williams * from entirely different assumptions, and can be tested with the help of Williams’s calculations of the measurements of Titoff, Homfray, and Chappuis. Reducing the equation to his units (adsorption a in c.c. of gas at N.T.P., pressures in em. of mercury), we obtain log « = log (2°988 . 108. £) —0°4348 . “ a = A )— Ay. . (14) The agreement found is very good, as is shown in the following table :— "PARR | Number of Observations. Pitrogen (Vitel) <3... .....<..00.5.-| 8 Methane (Homfray) ............... | 8 | Carbon monoxide (Homfray) Z| 10 | Oarbon dioxide (Chappuis) up to} : | an absorption of 40 per cent. / saturation, (excluding one EI dn io nwt ont dese, benawaes 49 a ee | | Mean divergence of observed | results from those calculated by “‘exponen- tial formula.” by equation (14). | | 0°9 per cent. | 7:0 per cerit. OG 5; [44 ,, DSO ret aam | | 0:8 ry) | ae Above 40 per cent. saturation, as might be expected, the approximation (14) ceases to hold exactly. I have also applied relation (14) to some measurements of Schmidt + on the adsorption of vapours of charcoal ;. these observations are not very precise (they do not lie evenly on any smooth curve), but for fairly low concentrations reasonable agreement is found. In fig. 1, log «/p is plotted against « for three series. From the experimentally determined values of the constants Ay and A,, values can be obtained for €, a, and No. These are shown in Table II. The values for a are positive small quantities, and, moreover, for the four gases the relative magnitudes are as would be expected, small for the inactive nitrogen, which cannot saturate much of the field of force, intermediate for methane and carbon dioxide, and largest for the unsaturated carbon monoxide. The values for &, which is a measure of the relative stability of the molecules on the surface, also follow the same sequence, the inactive gases having the shortest life on the surface. Finally, the * A. M. Williams, Proc. Roy. Soc. A. xevi. p. 287 (1919). + Schmidsz, Zeit. f. Phys. Chem. xci. p. 115 (1916). 694 Mr. D. C. Henry on a values obtained for Ny per gram of charcoal are of the same, and that a reasonable, order of magnitude. iow: : Log ¢ 1:0 © (t) 0-5 oe 3 4 Ob me EE GEO) 3 4 d 6 (2) 3 4 5 ge) Schmidt's Observations on the Adsorption of Vapours, (1) Carbon disulphide. (2) Benzene. (8) Chloroform, Yasue II. | | : N, Gas. T° C., Observer. a. pergm. per gm. | charcoal. charcoal. INMIUGTROTEE, | Wee eenooc Oo Raitott WeHO. |SU2Z5105 §\ 8-22 scites WMiethhamey 2 25.2 sod: 0 | Homfray.) 372 | 489x10-9 71D aos Carbon Dioxide...... | 0 | Chappuis | 3°32 | 1:47x107° 6°04 x 10° Carbon Monoxide...| —82 Homfray 474 | 317x107° 5:37x10°° Benzene. cos. a8. ccs. 6 | -15 Schmidt | 22 60s er the transformation (4) involves a velocity greater than that of light for the observer. To get over the difficulty we might use in this case the strictly legitimate transformation | Le tan 0.;= iz hy to a re collinearity is again produced, but now along the y-axis. There is. how- ever, no need to deal with it separately, it can be included with the other in the transformation (4) by imagining that the Lorentz equations are valid for values ot v greater than 1. It may be noted here that if we apply transformations (4) and (4a) to the same field, while (4) gives collinearity along « with er’ =H, h;’=H, (4a) gives collinearity along y with ey’=—7H, hy'=+7H. It will be evident in § 5 that these are the yz and 21 components of the electromagnetic five-vector (R, 7R), and that the two cases consequently form merely different aspects of the same field. ; Electromagnetic Lines and Tubes. 709 constructing a line of force in an electrostatic field, and the lines so traced out may be ealled ‘‘ electromagnetic lines.”’ Close to the origin the line passing through it lies along the w-direction satisfyi ing equation (1) in the arbitrary hyperplane in which the field is initially specified. This is the case because the rotation of the axes in the plane + (E+ H) (Se + £2) 20, hs mee Hoe. (E? +H) (Se + oe eee Ca) eee Hoey (E? + H?) C: om) = Ohta (al) | “A Boe o + i(B? + H?) = - oa) = 0,27 (t!) | po aS +i +B) (Se 4 Oe) =0,.. @) — = + i(E? + H?) (S24 a)=0, oc) Boe a (HP + H?) (Qe4 oe) = 0... @') §3. A Flux Theorem for each of four Electromagnetic — Lubes. To interpret the equations (8) we observe that through any point of the four-dimensional field not one only but jour electromagnetic lines may be constructed, each of which is uniquely determined when the initial hyperplane in which the field is specified i is given. Starting from this hyperplane the four axes’ directions required to produce collinearity at a given point are uniquely fixed. We can thus proceed to a neighbouring point lying on any of these axes, and rotate the system to. give collinearity. at this point, and ‘by con- tinuing this process obtain a continuous curved line in hyperspace which at the origin coincides with the given axis. We will call these curved lines the 2, y, z, /-electro- magnetic lines respectively; botnded by a set of each of them an &, y, 2, l-electromagnetic tube can be constructed in the usual way. Consider now an infinitely thin a#-tube, which we may take as having a rectangular cross-section at the origin. AZ Prof. 8. R. Milner on Let the adjacent edges of the section which contain the origin be infinitesimal lengths OY=y,, OZ=z (fig. 2) of the y and ¢-lines through the origin respectively, and let YY’, ZZ' be the 2-lines through Y and Z. The area of the section at the origin is y, 2, The cross-section at an infi- nitesimal distance along the z-line of OO'=a, will be a quadrilateral whose sides are ) 2 OY = nts, O'Z! = Atta ae and which will be altered in size, shape, and orientation from the original rectangle. If Y; and Z, are the projections of Y and Z by lines parallel to OO’, and we draw Y’A, Z/B perpendicular to O'Y, and O’Z,, we can readily see that 004 ZNiYA_ 1 id Jom OY oo YN , vce vy "1 (Oude and similarly Gd. Be ol Oz 21 O02 On substituting these in (8a) that equation becomes OF OH pay 8 (ye) =0 mie ct a or & ¢ JERE? . 4,2,) 00 Electromagnetic Lines and Tubes. (pk, This equation expresses the fact that the flux of the invariant function E?+H* over the cross-section 7,2, of an infinitely thin #-tube is constant throughout the whole tube ; since any tube may be formed by the juxtaposition of infinitely thin elementary tubes, the statement is true for a tube of finite cross-section also. In this case the corre- sponding cross-section may be defined as any continuous surface at any point of which the yz plane corresponding to the x-line passing through the point differs only infinitesi- mally from being tangential *. On the rectangle y;2; as base not only an w-tube but also an /-tube may be constructed, since the /- as well as the w-lines are at right angles to yz. The same figure 2 with OO’ representing the /-axis gives Oly mS 1 On 081; ak Oy mm Ol’ aes h and equation (80) reduces to a1 Ol’ co) s( AB ee Oe hd x) x,--C2 6) so that the flux theorem applies to the /-tube also. Any infinitely thin tube constructed of either y- or 2-lines will at the origin be perpendicular to the plane xl. Take the cross-section at the origin to be rectangular, formed of infinitely short z- and /-lines, 7; and 1,. We find as before Bye _ 13% yw _12h Of. Fie Ol lL, oy’ Bee _ 19m — 01_1 dh Ox 4 02’ Gio. Oa * In the general field the x-lines are twisted in the yz plane (v. infra) and a closed surface everywhere perpendicular to them cannot be uniquely constructed; 2. ¢., if we go from the origin always’ perpen- dicular to the «-lines distances, first y, and then z,, we do not arrive at the same point of the final z-line as will be reached by going the same distances in the opposite order, mathematically the conditions of uncon- ditional integrability will not be satisfied for the y; and z, displacements. For the purpose of reckoning the flux this feature of the tubes is immaterial, all the surfaces formed by joining up the points obtained by displacements in any order will over an infinitesimal region only differ from each other in area by second order quantities. The curvatures of the «-lines, and the second order displacements which they have under- oP at O', Y and Z out of the hyperplane zyz, are also of negligible eitect. 714 Prof. 8. R. Milner on With these substitutions (8c) and (8d) become S( VEF+HS wh) = 0, ae 2. ( YEE HE. al) = 0. . a It thus appears that a theorem expressing the constancy of the flux over the cross-section of the tube is derivable for each one of the four electromagnetic tubes which can be constructed with any point of Te field as origin. The cross- section over which the flux is reckoned is determined by the particular hyperplane which contains the infinitesimal portion of the tube concerned. The z- and /-tubes lie initially in the hyperplanes «xyz, /ye respectively, and have the same cross- section y;2,; the y- and ¢-tubes lie in yal, zwl, and have the same cross-section yl, The same quantity H?+ H? appears in each case as the function whose flux is constant ; it will be convenient to represent it by a single symbol R. In terms‘of the invariants of the field as usually expressed we have by (6) R= VB? = { ?— 7)? 4 ARP oe 5 (e a +4(eh)?}#. 5. (10) In the special case when e and h are Sere at right angles to each other (eh) = 0 and R = Ve? —h?. This is a result which has already been given in Prof. Whit- taker’s paper. The Lorentz transformation also shows that when e and h are perpendicular, H = 0 and R = H, hence when they are viewed at any point in. the appropriate hyper- plane the electromagnetic w#-lines are lines of pure electric force, they may however differ from ordinary electrostatic lines by their whole lengths not being containable in a single hyperpiane. § 4. A Theorem, complementary to the preceding, relating to the Twist of each Tube. The expression of the constancy of the flux of R for the four tubes only accounts for half the information derivable from the eight electromagnetic equations. The second four of the equations in (8) are concerned with what we may call \ 2 Electromagnetic Lines and Tubes. 715 the “internal composition” of the vector R. An electro- static tube in three-dimensional space is characterized by only two quantities at each point: its direction, and the electric force e which is inversely proportional to its sectional area at the point. An electromagnetic tube in hyperspace has not two but three such characteristic quantities. Not only will the same tube in two different places differ in direction, and in the magnitude of the vector function of position R, which latter is inversely proportional to the sectional area of the tube, but R= V7 EH?4 HH? will differ in the relative proportions of the collinear EK and H of which it is composed. The composition of R is adequately expressed by a parameter « such that hie H artis ey Sigh og a) bate Ce lac sina measures at any point the relative amount of magnetic force in R which cannot possibly be transformed away. We have further from (6) - ae 2EHH 2 (eh) sin 2a >= — — ——. EK? + H? R? Thus also sin 24 gives a direct measure at each point of the departure of the field from orthogonality of e and h, as expressed by the value relative to R? of the scalar-product ) I invariant 2(eh). Differentiating (11) we have EdH—Hdb = Reda, the substitution of which in (8 a'-d') gives Be 4 (Be oe) = 0, | Ow Oy F siti(S 2+ 52)=0 | ‘ : - (12) Ox . (0921 O08 x2 a. f Remy oe ray yaa si) =0, a2 : Oly obs) hac | Be (42 . Bese) _ 0, | These equations disclose that the variation of in each of the four coordinate directions is determined by the extent to a eR 716 Prot. S. R. Milner on which twisting occurs in a corresponding tube. In a twisted tube the bounding lines will not be parallel to the tube axis but will tend to run in spiral curves around it. Fig. 2 (p. 712) shows the effect of a twist on the wx-tube. Let oo — be the angles Y’O'Y,, Z'O'Z, through which O'Y’, O'Z' have been rotated from OY, OZ in the plane ye, while Y'YA, Z’ZB are the angles nse, oe through which the #-lines through Y and Z have been rotated in the planes zz, wy respectively from the z-line through O. We see at once by expressing AY’, BZ’ in terms of these that OO zx ie Oz, Oy O dz; uM Oy =r Ox >) 5. 5 aie ‘ : . (13) When Y'O'Y,, Z'O'Z, are not equal there is distortion as well as twist; the two may be separated in the usual way by writing byz= — bey = — 3 pzy' + bzy'')s Nom (14) Vyz= Vi —3( dey’ oe bzy'') measures the pure twist, or the rate at which the eos Mes yi, is undergoing rotation in its own plane in the direction from y to zas we proceed along the «-tube, Oye Ou while expresses the rate at which it undergoes dis- tortion. These considerations are applicable to any tube. For the z- and J-tubes erected on the same base W219 Obyze Ody Ot7e el z-tubes, based on 4l,, are twisted in the al plane ; let Odzi Ode 10702 (13) and (14), and corresponding formule for the other tubes represent the respective twists. The y- and be their respective twists. We then have from 00:2 OFyt _ r Odyz 0621 OO xz ee ¢ Oda Oy ar So ys =+2 Ais ae Al =+42 Bei Obzx 4 OFay _ _ 9 Obye EU ie OF ye _ _ 4 OGut OY Z OL’ Of) Ol Oye Electromagnetic Lines and Tubes. 717 and equations (12) become 222i Beg Bh | Ow ol OY Oe (15) Peas Ad as om) per sche Om _ 4 95 Oba? fo}! Ow Oz OY These equations, one for each tube, are complementary to the corresponding equations expressing the flux theorems, and show that the tubes give a complete representation of the electromagnetic field. Hquations (9) show that the tubes, by the variation of their cross-sections, determine the variations of R, equations (15) show that by their twists, they determine the variations of a, along each of the four coordinate directions. It should be noted, however, that while the variation of R given by each tube—aza, l, y, -— is that along the length of the tube, the variation of « given by the twist is along a direction perpendicular to the tube, but specially associated with it,—l, a, z, y. §5. An Electromagnetic Five-Vector characteristic of the Field. The meaning of these results can be made clearer by observing that the geometrical construction by which the electromagnetic lines passing through a given point have been obtained still leaves their directions in hyperspace subject to a certain amount of arbitrariness. They are only uniquely fixed when the arbitrary initial hyperplane is given (cf. § 3). In fact the final directions of the lines can be got from an arbitrary initial set of axes by four successive rota- tions in the planes of xy, wz, yl, and zl, and rotations in the planes zl and yz are not required to produce collinearity of e and h. They may be made of course, but they do not affect the collinearity. Thus in a hyperplane in which col- linearity is observed at a given point, let the observer suppose that he is in motion with a velocity v along the a-axis, zt. e. along the direction of EH and H. The Lorentz transfor- mation shows that this will affect in no way the magnitudes of E and H at the point; it follows, therefore, that the observer has no means of ascertaining his velocity in this direction, or of concluding that he has none, by observation of the field at this single point. We have therefore no right to assume that this velocity is zero, but in a general theory should write it as an arbitrarily given quantity. The corre- sponding transformation is equivalent to rotating the axes through an arbitrary angle in the plane of w/, and thus to 718 Prof. S. R. Milner on laying down in hyperspace a/- and J'-lines.in a different direction from before, although still in the unaltered al plane. Similar considerations apply to the y and z axes; indeed, when collinearity of e and h has been obtained it is evident that there is nothing to distinguish their actual directions in the fixed plane yz. | It thus appears that what is uniquely fixed in hyperspace is not the directions of the individual axes, but the orienta- tions at the point of the two coordinate planes al and yz, in which planes the axes themselves may be drawn arbitrarily. The entity that we have really to deal with in a four-dimen- sional theory of the electromagnetic field is of the type which is best represented not by lines but by surfaces; in other words, it is not of the four-vector, but of the six-vector type, being a function of position the properties of which at any point are associated with two absclutely orthogonal planes (vl and yz) which cut each other only at that point *. The six-vector in question is of a restricted type, characterized by the equality of its two parts; it consists of R= Vv H?+ H? associated with the yz plane by ‘‘ acting” in any direction per- pendicular to it, 2. e., along any line in the plane al, combined with the equal (but imaginary) quantity 7R similarly asso- ciated with the al plane. Although-a six-vector, since its two parts are equal, it. only requires five independent quantities to specify it: the four quantities required to fix the orientations of the planes, along with the magnitude of R. We will eall it consequently a “ five-vector,” thereby distinguishing it from the electromagnetic six- vector (h~ ze). The six quantities required to specify com- pletely the field at a point are known when a is given:in addition. . RES § 6, The Construction of Unique Sets of Tubes. _ These considerations enable us to explain the special sym- metry which the tubes show in the planes al and yz, and at the same time to derive a set of tubes which are really uniquely laid down in hyperspace. Having obtained from any initial hyperplane the axes oriented so that lies along the collinear E and H, let the axes be further rotated through the angle 0,7 in the plane wl, This will not affect E, H, or the axes y, 2. Starting from the origin along the new direc- tion w’ of w, we can construct an a’-line just as before by - * The plane «/ comprises all points of hyperspace for which y=0, z=0; yz all points for which «=0, 7=0; the two planes thus cut only at the origin. Electromagnetic Lines and Tubes. 719 infinitesimal rotations in the planes wy, a'z, yl’, zl! alone. a ; For the corresponding w'-tube we shall have fe sa era V KE? + H? ; Y121) = (cos Cis 2 + sin 0.7. =) ( / B+ H? . 21) = iy (oO aeeb). Hence the flux theorem applies to a tube starting from the origin in any direction in the plane of xl, the properties of the w- and /-tubes are in fact symmetrical in this plane. Since @,7 is arbitrary, we may choose it so as to satisfy any stated condition, for example, the condition that ; 24 = cos 0.192 + sin a = It is clear that in this way an 2’-line may be drawn such that there is no change in the composition of R along it, and it is now also uniquely laid down in hyperspace. When & =0, (15) shows that Oo = 0, which means that there is no twist round the /' direction perpendicular to z#'. Hence the “twist of the al plane,” 7. e¢. the twist round its axis of any infinitely thin tube drawn in it, is a maximum round the particular '-line which does not vary in composition along its length, while the l’-line at right angles to z' is charac- terized by no twist round it, and along it a maximum rate of change of composition. In fact, in the al plane (and all these conclusions apply also to the plane yz) the vectors representing the twist and the gradient of « are in mutually perpendicular directions *. 0. * Some sort of a visualization of the effects of twist can be got by picturing an z-line as being something like a ribbon instead of being the same all round like an ordinary line. We can suppose it shows E on the face and H on the edge of the ribbon. The Lorentz transformation enables one to alter the view point, and with untwisted ribbons to change the h-aspect of the field completely into e; when they are twisted however it is impossible to find any view point from which the e- or the h-aspect alone may be seen; they must both show simul- taneously. This agrees with what has been deduced: when e and hare perpendicular, one of them may be transformed away, when they are not perpendicular the lines are twisted and neither can be transformed away completely. It also enables us to visualize how twisted tubes might produce space-time variations in the ratio of H to E. I[t is, however, only a crude analogy and must not be pushed too far. Asa fact the composition of the lines changes in a direction perpendicular to the axis of twist, and not along it as the analogy would suggest. 720 Mr. A. Bramley on Radiation. Summary. When viewed in a suitable hyperplane, 2. e., when suitably transformed, the electric and magnetic forces at any point of the general electromagnetic field can be made to coincide in direction. This direction determines an electromagnetic line, continuous through hyperspace, from a set of which an electromagnetic tube can be constructed. Four such tubes, mutually perpendicular, can be constructed containing any point, and each is characterized by the constancy of the flux of the quantity R={(e?—h’)?+4(eh)?}s. over its cross section. A complementary theorem for each tube relating to the twist of its generating lines determines the internal constitution of R as expressed by the ratio of the magnetic to the electric force present in the compound vector. R is shown to be a “ five-vector,’”’ 7. e., a six-vector with its two parts equal; it is a function of position asso- ciated with two absolutely orthogonal planes uniquely fixed at each point of the four-dimensional field. The University, Sheffield, 29th June, 1922. LXIf. Radiation. By AntHurR BRAMLEY. To the Editors of the Philosophical Magazine. GENTLEMEN, [ the following discussion we shall make use of the idea that energy possesses mass, a principle which has been so fruitful in explaining the behaviour of light-rays in a gravitationai field and which is a natural consequence of the electromagnetic theory. If energy possesses mass, then the fundamental laws of mechanics ought to apply to it. Following this idea, we shall attempt to show how the laws of radiation are related to the fundamental principles of mechanics and electro- dynamics. The values of the potentials are : 1 0° Vier a ee 5? 1 9?U V Dicks taf U c2 0! Pp e” 10U Mr. A. Bramley on Radiation. 721 We shall suppose that the charged system is rotating in the XY plane around the origin as centre with uniform circular motion. Transforming to coordinates moving with the charge we have «=rcos ot, y=rsin at, or calling ot =0 we have in oe ne at oe But =o Sy, oe BG ERLE La eB, Baw let 6,6) I= 8 since the terms involving the velocities are negligible in comparison with the other terms. Now 0 7.0, ey = 0 (0419?) = 3 (1++2)o, dt 00°dt 00 dt/ 00 Ba (14 Bars 2, (222 4.2%) If 280 +108 =i. or = eae which is very probably the case for small oscillations then 0 +192 mo kt But the time during which the electron is oscillating 1s so great that kt-'< See ee oa C2 eile Cc += )88, a ae C2 V1 B? 1 Ge 00 e Further _10U:_2Uy__pd¢__ =8 24 r 00 fol) or a7 ee or” Hy= Us _ BU y a: St i ln NE GIR Lie ee ee Mr. A. Bramley on /tadiation. 723 Bey Oe in OP ei 08 eee eae pee eee OF a/ 1—f? Z ro / 1 — R? 00 pane Or Tah, Ose Pee a areal AC RAL OPS Now for the contractile electron the potential ¢, is symmetrical in the distorted space of the fixed system and H;= é e . ? equal to —— where 7, is the distance from the centre of Arr, the sphere in this system. P(xyz) O Rk Let P(ayz) be a point in space in the XY plane, 7) the projection of 7; on that plane, and 6; the angle POE where E is the position of the electron whose orbital radius is R. Then we have rere ters a rere 22, roe =r? + KR? —2rR cos 6, rear?+R?+27?—2rR cos 6). Therefore | = (7° + R42? —2rR cos 6,)'”, Sure : 1 dr a/7?-+ RP a?—DrR 03 By Whence we have which gives q, Sills caro Wamuate Dokl Lito oh Le: Oz (9? + R?—2rR cos 6, +27)?" dar” Odi _ —r+Reos9, Ou Or (7° + R?—2rR cos 0, 4+ 2”)9?" 4a’ Od1 _ —rhsin 8, e Oy int (7? + R2—2rR cos 0, + 2? i? * Aa’ ee Go 724 Mr. A. Bramley on Radiation. Then : Tis CO gia
) + ee (cos ae ) (6) * ‘This expression may be derived by remembering that if Ti, Uo, V3 are three orthogonal co-ordinates of such a kind that the differentials of length corresponding to the three differentials dij, dz, dads are alg O25 fre On a then . Vv 27 — O hy OV fe) ie 3 OS UR se (;. hs Sa) “O22 Ur hs ae. wn (2) See W. E. Byerly, ‘Fourier Series and Spherical Harmonics,’ p. 289, equation (6). Capacity of a Pancake Coil. 733 It is advantageous to transform this by = gin t. eke oe. Pa (0) y= 7 sinh;w, which reduces (6) to pee anya yO Mei Oe OV vV= 24a Be +2 {a ge (8) Now the electrostatic problem to be solved is that of finding for V a solution which together with its first derivatives is finite and continuous, which is independent of 6, which satisfies the Laplacian ete a wis Sais ut (O) which vanishes at infinity at least to the first order and which at the disk becomes V = Vo—L But the equation of the disk is i ee (Ns r di ei: (10) in which case (3) reduces to T= 0. COS\U, Hence, using (7), equation (10) becomes V = V.-Ld =p) %. Now the expression [anPn(v) + Bnldn(v) | [anPr() + bnQn(m) J, where P,,, Q, are Legendre functions of the first and second kind respectively, when substituted in (9) satisfies (9) in virtue of (8). If, then, one should be able to find such values of an, Bry Gn, 6,, and such values of n that V = > [ onPr(v) + BrQy(¥) | [ anP, (we) =F brQn (u) | should vanish at infinity to the first order and should degenerate into (11) when v approaches zero along the axis of pure imaginaries, then, in virtue of the uniqueness of the solution of (9) for given boundary conditions, the summation written gives the value of V. If the summation written is an infinite series it also gives V, provided it is universally convergent as to pw and v. (11) 734 Mr. G. Breit on the Effective Further, for a given vy, V may be represented by a series of the form ~ > AnPn(u); n=0 the summation being taken over all positive integral values of n, because V obviously satisfies the conditions which make such an expansion legitimate. ‘The coefficient A, is inde- pendent of w but, for different vaines of v, varies and is thus a function of vy. It must be clearly of the form iSuy = atnPn(v) ne B,QiV), for otherwise (9) would not be satisfied. Here n is a positive integer. The function P,(v) is therefore a finite polynominal *, viz. t. See en =) eee or na —in=2)\ 3) pW and (),(v) is an infinite series when | v/>1, viz. han ie LODE rere 1 (n+1)(n+2) 1 Qa) = eB be. nee Bie 2IOn+3) ven (n+ V(n+2)m4+3mt4) Vo 2.4(2n+3)(2n+5) Sito ee oe But points at infinite distance from the origin are given by real, positive, infinitely large values of u, and con- sequently in accordance with (7) by infinitely large values of v on the positive half of the axis of pure imaginaries. Such values can be denoted as usual by +j«. It is clear that if n>0, the expression for P,(v) becomes infinite for v= +) because it is a sum of terms of the same sign, and each term becomes infinite. Hence, if n<0; «,=0. Further, there is symmetry about the plane z=0. Hence by (3) and (7) only even values of n can be taken. Thus the most genera] possible expression for V is V = 2 don Pon(H) Qon(v). c : : (1 2) The coefficients A, as, must now be determined in such a way that ro) y aa © aan Pon(H) Qealj 0) = Vo-LUL— yp). . (13) n=() 1 Mise La es Pe * See W. E. Byerly, ‘Fourier Series and Spherical Harmonics,’ p. 145, equations (9) and (10). Capacity of a Pancake Coil. Top where the symbol Q»,(j.0) stands for the limit of Qon(j).5) as 6 approaches zero taking only real and positive values. Since now 1—p? = 3[ Po(w)—P2(w)], all do, but ap, a vanish, and 40» ad) are determined by the relations la ay Qo (7 - 9) = Vo—3L<, | ( 14 ee Ao Q.()- 0) — 2L ate | Thus li Vo—3L5 dt es Q.(j «sinh w) V= sinh w) +2 Lo z le ; Qj .0) Qo ) Q(z. 0) CH) } (15) The surface density of charge in coulombs is obtained as epueigeh 25) ey See ~ 8989 4 An’ where K_ is the dielectric constant of the medium, and a is the directional derivative of V with respect to the normal drawn away trom the surface. The same may be written as ee Am On’ ; where ga Coo) i yee Now at the disk the normal is parallel to OZ. Hence oT =(S)_,if 20 ang ee -) if 2<0, On Oz 2z—0 7 foe z=0 _ Hence by (3) and by (15), (16) pu J Qo (J - 9) 0) ~ seein YZ) G50 oy di 7 Q,' tJ 0) +3L5, oe yy Pl) } 036 Mr. G. Breit on the Effective But it may easily be shown that * psa) 20) ne Qo(j .0) ce and j Qe! (7 - 0) ae) CGO) ar Hence oe oa ee LS )+3 ah Pa(u) \ This solves the electrostatic problem proposed. Computation of the Function M(2). In order to find the function M(z), a choice must be made of the variable x Here r will be chosen as this variable. It was assumed that the e.m.f. varies as 7”. This means that r y2 {Moy qr= 15: or, differentiating, ip Mr) 2L5. . . * These formulas can be derived from using the following facts :— (2) Qa(z) = Flog 25, Pg—] I Ole gag 7% Ont DP, WAY) (ec) The recurrence formulas (2n+1)zP,,(z) = (wn F1)P,, 41 (2)+nP,,_1(2). Using (c) in (0), the expression for Q,,, in terms of Q. can be derived by writing the identity Then from (a) expressions for Qo,, are obtained. On differentiating these expressions and passing the limit in the result of the differentiation as well as the original, the result follows at once. Capacity of a Pancake Coil. 737 Computation of the Function a(x). As stated in the introduction, the differential of charge is 2 a(x). da. The independent variable here chosen is 7. As? varies from 0 to a, and as @ varies from 0 to 27, the whole coil is traversed by the point (7, 0). The differential of area is 27rdr, and the differential of charge is then 4aordr, where o is given by (17) becau-e expression (17) gives the surface density only on one side of the coil. By (3), on the coil » becomes acosv; so that na —— /1—cos?v = /1 = Substituting this in (3), and expressing the fact that di 4rordr = (5) a(r)dr, it is found that A= Vea) (ay 3) SPV)? on (\(a)= x It now remains to substitute (18), (19) into (1). If the ungrounded condenser terminal is connected to the centre, x, is to be taken as 0 and 2 is to be taken as a. IE, however, it is connected to the periphery, x, is to be taken as aand a, as 0. ~The first of these gives Bee P20: 20) and the second C: _ 4a (2 J a (21) jy =Biss eS a |. ek a Zu 7 [9 s Lo If, now, the coil should be used with the centre grounded, and the ungrounded terminal of the condenser should be connected to the periphery, formula (21) applies, and in that formula V)>=0. ‘This gives SKa 3 ; peewee ASAE [a TS ie ra Oy - Sean es (22) This is the effective capacity if the centre is grounded. Phil. Mag. 8S. 6. Vol. 44. No. 262. Oct. 1922. aB 738 Mr. G. Breit on the Effective Again, if the periphery is grounded and the centre is not, di formula (20) applies, and in that formula Vo=L et is seen from (10) by setting V=0 when r=a. This gives | 14a n= SS 2: Co loa 2 for the capacity with periphery grounded. Finally, if the coil is insulated and the condenser is unshielded, as much current enters the coil as leaves it ; so that (20) and (21) must give the same value for Cp. Multiplying (20) by 2 and adding to (21), it is found that _ 32Ka G= Fee. 6 er if the coil is ungrounded. It is worth mentioning that if Cy be eliminated from (20) and (21), it is found that ue which, in virtue of the Le 1 { P2(u)dp = 0, 0) shows in a different way that the coil is insulated. Expressing the results in micromicrotarads, the capacity 1s identity when grounded at centre.....;... 0°567 Ka ueuf, when grounded at periphery ... 0°330 Ka puyf, when insulated. =) 4)) ieee 0-252 Ka ppl. Now, according to the results cf a previous calculation *, the effective capacity of a pancake coil of small depth when insulated is 0°437 Ka. Thus, so far as the effective capacity is concerned, there is an advantage in using pancake coils of large depth as compared to pancake coils of small depth. EHaperimental Verification. The formulas (22), (23) have been verified experimentally on a coil which is shown in fig. 3. This coil is not cireular * See G. Breit, 2. c. Capacity of a Pancake Coil. fel but hexagonal. The quantity a is therefore not quite certain. In the computations it was taken as the mean of the radii of the inscribed and escribed circles, which are 26:5 ems. and 29°5 ems. respectively. Hence the mean is 28 cms. The dielectric being air, the capacity with centre grounded should be 16 mwuf, and if grounded at the periphery it should be 9 wuf. The values as measured are 16 put and 9 upl. Fig. 3._-Photograph of pancake coil. Measurements were not made, more accurately than 1 pf, on account of the difficulties connected with such measurements. The capacity of the same coil was also measured without the copper foil, leaving only the copper braid. No change was detected in the capacity. It also appeared that the copper rods used in fastening the braid could affect the capacity. A row of them was soldered to the braid, but no detectable change in capacity was noticed. 3B 2 740 Prof. Hackett on Relativity-Contraction in a Rotating Conclusion. The effective capacity of a pancake coil has heen calcu- lated, and the calculations have been verified experimentally in two cases. The results of the onleulation are that the capacity of the coil when grounded at centre is ......... 0°567 Ka wpt, when grounded at periphery is ... 0°330 Ka ppf, when insulated is ............ .. ila ee 0°252 Ka pp, where a is the radius of the coil and K is the dielectric constant of the medium. Washington, D.C.. Jan. 14, 1922. LXIV. The Relativity-Contraction in a Rotating Shaft moving with Uniform Speed along its Avis. By VWurrx HW. Hackert, M.A., Ph.D., Professor of Physwes, ‘College of Science for Ireland, Dublin * § L. Introduction and Summary. SOLUTION is offered in this paper of the problem of the relativity-contraction in a rotating shaft moving with uniform velocity along its own axis. The standpoint adopted is that of the fixed ether and the FitzGerald-Lorentz contraction combined with the restricted principle of rela- tivity. ‘The validity of Huclidean geometry is assumed throughout the paper. A hypothetical modification of Fizeau’s method for measuring the velocity of light is considered—a rotating shaft carrying two disks with apertures which correspond to the toothed wheel in Fizeau’s experiment. It follows readily that when a rotating shaft is moving with uniform velocity along its own axis, to a stationary observer, looking in the direction of motion, it appears twisted in the opposite sense to the rotation. This effect has been pointed out by R. W. Wood +, and he has discussed the experiment, but not in a sufficiently precise way to serve as a basis for the subsequent discussion in this paper. The arrangement may act asaclock. It measures time * Communicated by the Author. Read at the meeting of the British Association, September 1921. + Wood, ‘Physical Optics,’ 2nd edit. p. 690. Shaft moving with Uniform Speed along its Avis. 741 on the same principle as the ideal clock consisting of a beam of light reflected between two mirrors, with the addition that a disk fixed on the shaft at any cross-section and rotating with it can, owing to the twist, indicate the local time there. In the latter p: os of the paper, the contraction in the shaft due to the motion of translation and the twist is considered as a strain-displacement. One of the principal axes of the strain is assumed to be the direction of resultant velocity 7? 4 wu. The principal contraction in the latter direction is found to be W1— (v?-+u2)/c?. This result holds for a shaft of any ferm, since the twist and longitudinal contraction do not depend on the form of the oath Passing from the case of a solid circular cylinder to the limiting case of a disk rotating without any motion of translation, the reasoning in this paper gives the circum- ferential contraction as equal to that usually accepted for a rotating ring, viz. W1—vw?/c?, where wu is the velocity at the rim, Tt follows that the conichion in the radius is vee the same magnitude. § 2. The Velocity of Light and a Rotating Shaft. Stationary System.—A rotating shaft can serve in theory for the determination of the velocity of light by the following modification of Fizeau’s experiment. Two similar disks are mounted on the shaft in planes normal to the axis separated by a distance 1. Hach disk is perforated by a number of equidistant apertures lying on a circle concentric with the shaft. In the subsequent discussion we ignore ordinary elastic strains, or, in other words, assume that the elastic constants are infinite. The axis of the shaft is taken as the axis of z. The disks are similarly placed so that the apertures in each disk pass simultaneously through the plane of («, z) as the shaft turns. We need only consider light rays travelling in this plane parallel to the axis of the shaft so that they can pass through an aperture in each disk for suitable speeds of rotation. Let the period of the lowest of these speeds be T. For this speed, light travelling through an aperture in one disk to- wards the other will pass through another aperture there which has just been brought into position by a rotation of the shaft through an angle ¢, where ¢=angle between two successive apertures in either disk. We have then eat. ee. - at) : 742 “Prof. Hackett on Relativity-Contraction in a Rotating Moving System.—On the theory of restricted relativity, if the apparatus is transferred to a system 8’ moving with speed v along the axis of z and parallel to the axis of the shaft, the properties of the system remain unaltered. For the moving observer there are, therefore, definite speeds of rotation for which (1) light rays parallel to the axis of the shaft can pass through the apertures in the disks in either direction, (2) the measured speed of light in this hypothetical experiment is c. To an observer in the stationary system, this is impossible unless compensations take place. He knows (3) the velocity of the moving system, (4) the distance between the two disks modified by motion to 1V1—v?/e?. Reasoning on these data, he concludes that the experiment can only succeed if the forward end of the shaft when in rotation is twisted, with respect to the rear end, in the opposite sense to the rotation through an angle, say @; and this twist must be such as to compensate for the different light-times between the disks in and opposite to the diree- tion of motion. During the light-time for the former direction the shaft turns through an angle 6+, while the light has the relative velocity c—v. In the other direction, the shaft running at the same speed turns through an angle @—O, but the ight has the relative velocity e+v. At this speed, condition (1) is satisfied, since light, emitted through an aperture in one disk towards the other, reaches it just as an aperture is passing across the path of the ray in the plane of (,2). The period of rotation T’ and the twist required to satisfy this condition can be determined by the fixed observer using his own units from the equations :— (c—v) (6 +0) T/20 =1 V1 —v?/e? = (c +0) (6—8) T'/27, which give | CHGS Og b.¢. (1—v/e?) Waa = 1 103 /e?. Shaft moving with Uniform Speed along its Aais. 743 Using (1) we get Eo So] re.) lv 20 luw = VS es = SSS (3) CV1L—we Th ae V1 For the moving observer, however, there is no twist in the shaft. His units of time and length have altered, so that equation (1) holds giving the speed of light as e. § 3. Lelativity Clocks. This combination of rotating shaft and disks may be regarded as a set of relativity clocks regulated by the property that the shaft must rotate with the slowest speed for which it can transmit light through the apertures in either direction. The fixed observer considers that such clocks in the moving system run slow according to (2). It will be shown below that there is an automatic synchroni- zation of the clocks. This is produced by the twist in the mechanical coupling, and satisfies Hinstein’s test for synchronism. In the simplest form, each disk can serve as a local clock. To give the same value toc, the same method of fixing the unit of time must be adopted in all systems. It is con- venient, here, to take as the unit the time of describing one radian. Using this unit, the angular position in radians of a special aperture with respect to (#, z) gives the time directly, and from (2) o = V1—v?/c? = angular velocity of S’ shaft in S-units. (4) To the moving observer there is no twist in the shaft. If he arranges that the timing aperture shall lie for every disk in the plane of (x, z) when the shaft is not rotating, he will conclude that in rotation the timing apertures pass through this plane “‘ simultaneously ”’. It is easily seen that Hinstein’s test for synchronism 1s satisfied. The first disk may be taken as the origin, and the second at a =/S/-units since 8’ ignores the contraction. Let all the apertures be numbered in the opposite sense to the rotation 0, 1, 2, 3, ete., bsginning from the timing aper- ture, and let mirrors be fitted into the apertures in the second disk. For the speed w, a ray leaving the first disk by No. 0 aperture is reflected at the second disk by No. 1 mirror at 2’ and returns through No. 2 aperture. Thus the time of 744 Prof. Hackett on Relativity-Contraction in a Rotating arrival at vw’ = mean of departure and arrival times at the origin. According to the reckoning of the fixed observer, however, the passage of the timing aperture at 2’ through ‘the plane of (xz) is from (3) later then at the origin by w’va/e? W1—v?/e? owing to the twist in the shaft. This is the local time eftect and leads to the Lorentz time-transformation adopting the » usual conventions. Counting time from the instant when the origins in each system coincide and the timing aperture at the origin in 8’ passes through the plane of (z,z), we have at every point along the shaft rotation of the 8’ shaft its angular velocity in S-units. fin S=time = The rotation of the S’ shaft is got by adding the twist to the angular distance of the timing aperture from the plane of (a, 2). The latter is ¢’ in the units we have adopted, and from (3) and (4) we have 0 = w’v/c?, giving WAR IEG EA 5: 1 — P ¢ in S-units = W=ae @ +0) — ie Wie ee 5 Cae tale Ne (5) § 4. The Strain in the Rotating Shaft. The discussion in §2 has shown that to a fixed observer a rotating shaft with a motion of translation in the direction of its own axis is in a state of strain. For convenience the term contraction is used here for the ratio of new length to original length. We shall now proceed to find the principal contractions for this state of strain. For the sake of clear- _ness, the assumptions involved in this discussion are set forth below. We have as data the following deductions from the appli- cation of the principle of Restricted Relativity to the systems considered in §2:— | (I.) The twist in the shaft is independent of its radius, whether it is hollow or solid, and is given by (8). CII.) The FitzGerald-Lorentz contraction of the distance between two planes perpendicular to the axis is not altered by the rotation, otherwise the relations deduced in (2), (3), and (5) in agreement with the ordinary theory could not exist. The contractions of the relativity theory are independent Shaft moving with Uniform Speed along its Avis. 745 of the physical characteristics of the body, and consequently, as already stated (§ 2), the fixed observer eliminates from his consideration any distortion due to centrifugal forces by the following condition :— Ordinary elastic strains are ignored or, in other words, it is assumed that the elastic constants are infinite. He interprets his observations in terms of Euclidean geometry, and so he makes the following geometrical assumptions :— (A) Each cross-section of the shaft remains a Huclidean plane, so that its radius alters in the same ratio as its circumference. Assumption (A) is the only possible one from the Euclidean standpoint. Its justification lies in the simple form in which the principal contractions appear below. Using these deductions and the foregoing assumption, it will be seen below that we can derive expressions for the contractions along the principal axes if we know their posi- tion. At this stage a further assumption must be made, more speculative than the preceding which arises directly from the Kuclidean point of view of the fixed observer. According to (A) one of the principal axes of strain lies along the radius, 7. e. along the direction of the centripetal acceleration. The other two must then lie in a plane normal to the radius, and we assume that (B) one of these lies along the direction of resultant velocity. This last assumption may be justified by analogy with the Wiedemann effect. To a fixed observer a rotating shaft in the form of a thin tube moving along its own axis will be in a state of strain similar to that of a steel tube placed ina coaxial spiral magnetic field. In this instance, if hysteresis be eliminated, as it can be by special experimental methods, one of the principal contractions must be along the ee aiaad magnetic field. The formula for the twist in the tube, deduced on this assumption, has been confirmed by experi- ment. In one case the tube is twisted by a spiral magnetic field, and in the other by its spiral motion. The physic: al analogy is so close that it seems to the author to justify the fixed observer in applying the same analysis to the twist in each case and in making assumption (B). about the direction of the principal axis. We shall now examine the state of strain in the shaft from the point of view detailed above. In considering the analogy 746 Prof. Hackett on Relativity-Contraction in a Rotating of the Wiedemann effect the shaft was taken as a thin tube; this restriction is not necessary for the general mathematical treatment, as can be seen by reference to deduction (1.), though it may be helpful to think of the shaft as a thin tube in the following discussion. We consider that the new co-ordinates 2’, y’, z' of any point, as interpreted by the fixed observer, are given in terms of the original co-ordinates as in the ordinary strain theory. The axis of the shaft is taken as the axis of z. The fixed observer infers a rotation 0’=7z according to (3), and a FitzGerald-Lorentz contraction parallel to ¢ according to deduction (II.). In accordance with assumption (A), each radius in any plane parallel to (wz, y) is assumed to be con- tracted in a ratio which depends only on 7, and the circum- ference alters in the same ratio. As the rotation @’ cannot be assumed generally to be small, the steps of deducing the strain-components are given below :— w' = excos 0’ —ey sin 0, y' = ex sin 0 + ey cos 6. We get for the relative displacement &', m’, & around &, Y, g, of a point whose undisplaced co-ordinates are w+&, yt, 246: Toe roy Oz! | ! gia £0 + 9 E— try oy’ Oy — - u e n= a ae 4) a + Cre But part of these relative displacements 1s a pure rota- tion 6’ around an axis through a’, y’, 2’ parallel to the axis of the cylinder and arising from the general rotation. We shall obtain the strain components &', 7’, ¢ by combining the total effect &,', n', &/ with a rotation —0’, where £'— £/ cos 0'+7n// sin 6, n' = —&/ sin 0'+/' cos 0. Whence a¢er) ie Cle pi EY 4 ee tray), 1= po + + Er(ee). Shaft moving with Uniform Speed along its Aais. 747 When we refer to a point «=r, y= 0, we get g'= E(e+r S$) + a chalga GO) CO — ent+err€, esc mee BACT ies Bier ter) fe (7) The radial strain at any point is determined by equa- tion (6). To find the principal axes of strain in the plane of (y, z) we need only consider equations (7) and (8), which may be written FCT ASE s ghee IE tee te ost eel as CD Cee Rape te es eye yo ee | ee CO.) where Se eNews at Nee ure, Ga) f= V1—v/e by deduction (II.). (2) The next step is to express e, /, s in terms of the position of the principal axes and their contractions. Since the strain is not pure, the principal axes will be rotated from their initial positions, and both their final and initial positions will have to be considered. Fig. 1. \ Final | ‘ JY position \ Y /attia/ pos/tion We assume this part of tle strain is produced by contrac- tions along and perpendicular to axes Y and Z, which in consequence of the strain have been rotated into positions y¥ and z, as shown in the diagram. 448 Prof. Hackett on Relativity-Contraction in a Rotating We have y'= py, 2 = 9K. The co-ordinates y, z and axes Y, Z introduced here have no reference to the co-ordinates and axes used to deduce equations (6), (7), (8). Returning to 7, € axes, we have n' = y' cos6—z' sin 6, C= y' sind +2' cos 6, Y= ncosa+€sin a, Li — sin «+ (cos 23. which give n' = (pcos « cos 6+ q sin & sin 8) + €(p sin « cos 6—g cos « sin 6), ¢’= (pcos a sin 6—g sin a cos 4) + €(p sin asin 6+ q cos & cos 6). Comparing with (9) and (10), we get e= pcosacosdot+qsinasiné, . . . (13) s=psinacosé—qcosesinéd, . . . (14) f=psinasind+qceosacosd, . . . (15) O=pcosasind—q sin xcos 0. :. . yay We get from (14), (15) fsin 64+ scos6 = psin«, f cos é—ssin 6 = 9 cosa, giving : = (f+s cot 5) a Saye acai (17) g = (f—s tan yee gt ah ele from (16) ee deine aE ee 12. OUT a gre a eae! Shaft moving with Uniform Speed along its Aats. 749 Thus pp? f+scotd ; g? f—stand’ - (20) ae SEEN ° e ° ° . ° ° “ (21) oe 9 ee bere eich 3 OD At this point assumption (B) is introduced, 2. e. one of the principal axes in the final position lies along the direction of resultant velocity, and we write an efor OE a Se le OO In the expression for the twist obtained in § 2 we have to note that 0’ is in the opposite sense to @, since translation is. in the positive direction along z, and we introduce a negative sign, writing 0 /l = —ar/? V1—v?/2 = 7; hence Serr wc’ 4/ Lose? 8. (2A) and from (12) Inserting these values in (20), we find pr Pe ale Cae al yo aa teed ge (2G) giving Geant — Uw A/a? 0? tle. eam, (27) and from (23) and (27) sin a = sin 6 /1—7/P? — w/e? |W 1—w se. . (28). Inserting these values in (19), (21), and (22), we find the values of e, p,and qg. The contractions are, most con- veniently, stated for the surface of a cylindrical shaft having a uniform motion of translation v and a rotational speed wu at the periphery, where both velocities are measured by the fixed observer. Symbol. Direction, Contraction. p Resultant velocity. V 1—v/? —v?/c?. e Circumferential. VIP —w/e2 [| f1—v?/c. Longitudinal. Vv 1—v"/c*, q Normal to acceleration I and resultant velocity. 750 Prof. Hackett on Relativity- Contraction in a Rotating The way in which the form for the simple longitudinal contraction is maintained in this more complex motion sup- ports the assumptions which we have made, especially assumption (B)}, giving in (23) the direction of one of the principal contractions. These results hold generally for any solid shaft in the same state of motion, since, as stated in deductions (I.) and (II.), §4, the twist and longitudinal contraction are independent of the form of the shaft. § 5. The Contraction in a Rotating Disk. The expressions deduced in the last section hold for all values of » and wu. They should hold in the limiting case for which v=0 when the shaft is rotating around an axis fixed relative to the observer. In this way we derive a solution of the problem of the rotating disk which enters so frequently into discussions of the restricted and general principles of relativity. Writing v=0, we find circumferential contraction = /1—uw?/c?. Before discussing this result, it may be well to state the solutions which have been previously given of this problem. Following Ehrenfest, it has been frequently stated * that if a measuring rod is applied tangentially to the edge of a disk in rotation in its own plane about its centre, the rod is shortened in the direction of motion, but will not experience a shortening if it is applied to the disk in the direction of the radius. The result was originally put forward as a speculative inference from the restricted principle of relativity. It raised the difficulty that the ratio of the circumference to the dia- meter is no longer constant, but this has since been met by the statement that the disk is no longer a Euclidean plane. On the other hand, Lorentz finds that both radius and circumference contract in the ratio of 1 to 1—v?/8c° from an investigation based on the general principle of relativity. This problem is a special case of the “ general question as to how far the dimensions of a solid body will be changed — when its parts have unequal velocities, when, for example, it has a rotation about a fixed axis. It is clear that in such a case the different parts of the body will by their interaction hinder each other in the tendency to contract to the amount determined by /1—v[c?”f. * Einstein, ‘Theory of Relativity,’ p. 81; Jeans, Proc. Roy. Soe. vol. 97. A, p. 68 (1920). t Lorentz, ‘Nature, February 17th, 1921, p. 79. Shaft moving with Uniform Speed along its Avis. 751 The statement just made does not seem to apply to the problem treated in this paper. For the contraction due to the helical motion of the shaft is given in terms of thie resultant velocity by the usual formula; and according to deduction (I.) in §4 the state of strain in the periphery of a solid shaft is in no way different from that of a thin tube of the same external radius and in the same state of motion. In the limiting case no distinction can, therefore, be drawn between the strain in the rim of a rotating disk and a rotating ring. ‘The radius of each must contract in the same ratio as the circumference, viz. in the ratio 1 to V 1—w?/c?. We have then rg=r V1—o?r,?/c?, where 7, = radius when the angular velocity is m ; this gives dy dy) (Ul — wr? cP. Summarizing these results, we have then :— The radius and the circumference of a solid disk rotating with constant speed about an axis at right angles to its plane contract in the ratio of 1 to W1—v?/c?, where wu is the velocity at the rim. A measuring rod laid along the radius contracts in the ratio of 1 to (1—1w,?/c?)??, where w, is the velocity at that position in the disk. The simplicity of the assumptions made and the analysis given in this paper give support to the view that the above conclusion is correct, within the limitations of the Euclidean outlook adopted. It takes a middle course between the results stated by Hinstein and Jeans and the solution given by Lorentz. It is conceivable that a solution may, however, be found beyond the limits of Euclidean geometry which may include all points of view. ee § 6. Note on the Wiedemann Effect. A vertical iron wire carrying a current twists in a vertical magnetic field. This is recognized as an effect of magneto- striction due to the resultant magnetic field in the wire. The effect is simplified if a steel tube is used in which a spiral, or more accurately a helical, magnetic field acts whose axis coincides with the axis of the tube. This is produced by combining a longitudinal field with a circular fieid due to a current flowing in a wire passing along the axis of the tube, cee 752 Dr. T. J. Baker on Breath Figures. The analysis given above immediately applies; we get from (11), (13), (14), and (19) O'/l = 7 = sfer = sin 26/r . p?— q?/2pq. In the case of magneto-striction we can write p=1+«¢ and g=1-+e,, where e, and e, are small. The difference between the position of the principal axis before and after the strain can be neglected so that 6= «a, and we get 6’ = sin 2a (€,—€) l/r. F This is the formula given by Knott and verified experi- mentally by the author *. The author desires to express his obligation to Prof. W. MeF. Orr, F.R.S., for his interest in and criticism of this paper. LXV. Breath Figures. By T. J. Baker, D.Sc. (Lond.)f. F one breathes upon a sheet of glass which has been cleaned with soap and water and polished with clean linen, water-vapour condenses uniformly on the glass in such i. manner that the surface as seen by reflected light appears dull and rather white. If the tip of a small blowpipe-flame is caused to traverse the surface of such a plate and the plate — is then breathed upon as soon as it is cold, a whitish con- densation appears on those parts which the flame has not touched, whilst the track of the flame is marked by a form | of condensation which, owing to its transparency, appears | black by contrast with the neighbouring parts. This and certain allied phenomena were described by Aitken ¢ in 1893, and several letters discussing the subject appeared in the pages of ‘ Nature’ § during the period 1911 to 1913, but no general agreement as to the cause was reached. | A lens shows that the white portion of the deposit consists of lens-shaped drops which are isolated from each other, whilst the black condensation consists of a continuous film of water. * Knott, Trans. Roy. Soc. Edin. vol. xxxv. p. 388; Hackett, Proc. Roy. Dub. Soc. vol. xv. (n.s.) p. 416. + Communicated by the Author. { Aitken, Proc. Roy. Soc. Edin. p, 94, 1893. § ‘Nature,’ May 25, June 15, July 6,1911, Dec. 19, 1912, Feb. 6719138. See also vol. vii. of Lord Rayieigh’s ‘ Collected Scientific Papers.’ Dr. T. J. Baker on Breath Figures. 133 The late Lord Rayleigh held the view that the part of the elass swept by the flame had been rendered cleaner than the neighbouring portion, whilst Aitken urged that the track of the flame had been rendered dusty by solid particles deposited from the flame, and that these particles aided condensation of moisture. He pointed out that by scraping with a match- stalk across the flame track a dusty deposit could be rubbed up, and he considered that this contamination of the surface is responsible for the breath figure. It is true that the track which an ordinary blowpipe-flame has followed can be detected by the eye before any moisture has been deposited on the plate, but a flame of carbon monoxide leaves no such deposit, and Aitken’s explanation seems inadequate because this flame yields an excellent breath figure. Lord Rayleigh showed that if the outside of a test-tube were heated to redness the “black” or transparent condensation could be obtained on the corresponding part of the interior of the tube. This was at first contested by Aitken, who maintained that the flame, or the hot gases from it, must strike the glass directly to produce the result ; but further experiments led him to accept Lord Rayleigh’s statement, and he then sug- gested that a chemical change in the glass itself might account for the effect. Quincke found that when a drop of strong sulphuric acid is warmed on a glass plate, which is afterwards washed and dried, the ‘“ black”? condensation may be obtained on the part which has been exposed to the acid. Craig suggested that this might be due to the soaking in of the acid, thus forming a hygroscopic film; and as coal-gas always contains sulphur compounds, he contended that a coal-gas flame ‘playing on glass might deposit enough suiphuric acid to act in this way. Butit was known that a flame of pure hydrogen burning in air also gives breath figures, and Craig therefore suggested the possibility that some nitric acid might be formed by the flame, and that this acid might determine the pro- duction of a breath figure. If this is true, it follows that hydrogen burning in pure oxygen should fail to be effective. This summary fairly represents the main features of our knowledge of the subject up to 1913. From that date onwards the author has intermittently carried on the inquiry, with the result that other relevant phenomena have been discovered, and a partial explanation of the results can be offered. Phil. Mag. 8.6. Vol. 44. No. 262. Oct. 1922. 30 Peo” T54 Dr. T. J. Baker on Breath Figures. Heperimental, With the exception of certain cases specifically mentioned, the surfaces used were first washed with soap and water, then rinsed with tap-water, dried with a clean cloth, and — finallyrubbed vigorously witha clean linen handkerchief until the white condensation produced by breathing lghtly upon them showed uniformity. ‘The surfaces so prepared will be described for convenience as ‘ cleaned” surfaces. It must also be understood that the term ‘‘ breath figure’ connotes that condition of a surface which reveals itself by the black form of condensation. A breath figure is not visible until moisture condenses on the surface. At the outset it seemed probable that a complex substance like glass was not best suited for the purpose, and some pre- liminary experiments were made with other materials. In general, it was found that the chemical composition of the substance is not important, because breath figures were easily obtained on porcelain, rock-crystal, mica, Iceland spar, platinum, nickel, silver, brass, and mercury. Aitken’s sug- gestion that chemical change might explain the effect cannot be maintained in face of the fact that the figures can be obtained on rock-crystal and platinum. Again, since the burning of coal-gas in a blowpipe-flame might lead to the deposition on a cold surface of such substances as carbon, sulphur compounds, and tarry matter, it was decided to try the flames of hydrogen and carbon monoxide. Hydrogen prepared from hydrochloric acid and magnesium was passed through soda-lime to arrest any acid spray, and was burnt at aplatinum jet. This flame produced excellent breath figures. Burning carbon monoxide from sodium formate and sulphuric acid gave equally good figures. From these experiments we may conclude that breath figures do not require for their production the deposition of any solid matter from the flame. Further, since no water is produced when carbon monoxide is burnt, the effect cannot be attributed to depo- sition of moisture. Lord Rayleigh has pointed out that the pattern of a breath figure may be recorded permanently by the chemical! depo- sition of silver on the glass plate. ‘The track of the flame is distinctly marked by a difference in the appearance of the deposit, and this difference is most marked near the margins of the track, i. e. where the hottest part of the flame impinges on the glass. This was confirmed both when the flames of hydrogen and coal-gas were employed. Dr. T. J. Baker on Breath Figures. 7155 Injluence of the temperature of the flame. Aitken mentions that the flame of burning alcohol does not produce satisfactory breath figures. The author made a small spirit-lamp with a test-tube as reservoir and a tuft of elass-wool supported in a glass tube as wick. This was supplied with *‘ industrial spirit,’ and was regulated to give a flame about + inch high. No breath figures could be obtained, probably because such a small flame is not very hot. It was found that the jarger flame of an ordinary spirit-lamp gave a breath figure, but the effect was distinctly weak. Some methylated ether was purified by standing over caustic potash and potassium permanganate for several days and then distilling. The portion coming over at 34°°5 was collected and used in the lamp described above. No breath fizures were obtained. It may be noted that an ether flame always leaves a small deposit of soot on the plate, and the non- production of a breath figure in these circumstances seems to negative Aitken’s suggestion that dust is an 1m- portant factor. ~ Ifa mouth-blowpipe is used in conjunction with an alcohol or with an ether flame good figures are easily obtained. Again, if a glass plate i is drawn rapidly across the extreme tip of a well- shaped blowpipe-flame, the breath figure shows a perfectly uniform “black” track ; but if the. plate cuts across the flame near to the red acing cone, then the breath figure shows two clear “ black” lines which correspond with the hot exterior of the flame, whilst the space between them exhibits more or less of the white condensation corresponding to the relatively cool interior of the flame. These facts seem to indicate that the condition of the surface necessary for the production of these figures is only attained after it has been exposed to a flame whose temperature is above a certain minimum. Sir J. J. Thomson (‘ Conduction of Electricity through Gases,’ p. 194) says “ionized gas is produced by flames ‘of coal-gis whether luminous or not, by the oxy-hydrogen flame, by the alcohol flame of a spirit- lamp, by a flame of Sarapnte oxide: it is not, however, produced in very low temperature flames such as the pale lambent flame of ether.” The parallelism between the ionizing effects.of the flames mentioned in the foregoing abstract and the facts just stated in connexion with the production of breath figures suggests 302 756 Dr. 'T. J. Baker on Breath Figures. — the possibility that ions derived from the flame may be the cause of these figures; but the extraordinary permanence of the effects and the fact that actual contact with the flame gases is not essential render the hypothesis untenable. Permanence of breath figures. The peculiar condition of the surface of glass which causes it to reveal the flame-track when it is breathed upon is singularly permanent, and persists for many months. As stated by Lord Rayleigh and Mr. Aitken, a breath figure may be removed by rubbing with soap and water, but the author has occasionally experienced great difficulty in getting rid of the last traces of the effect. Rubbing with a dry cloth weakens the figures, but does not destroy them. Huperiments with chemically cleaned glass. Up to this point the glass plates had been cleaned as described on page 754, and it now appeared necessary to examine the behaviour of glass which had been subjected to chemical cleansing processes such as are employed preparatory to silvering. After the final washing in distilled water the plates were supported on glass reds in a desiccator, and were left there until dry. | When these chemically clean plates were breathed upon the condensation was almost entirely of the “‘ black” kind—or, in other words, the glass was covered with a continuous film of water. When a flame was made to traverse a chemically clean plate and was afterwards breathed upon, no breath figure, or at most a very imperfect one, appeared, thus sug- gesting that a film of some contaminating material must be present upon the glass before it is exposed to the action of the flame. Glass plates which have been cleaned as described on page 754 are certainly covered with a contaminating film, and in this connexion the work of W. B. Hardy andJ. K. Hardy (Phil. Mag. July 1919) is significant. These investigators found that truly clean glass surfaces will not slide over each other, but seize owing to cohesion. A very small amount of contamination lowers the resistance to relative motion, and sliding becomes possible. The author has applied this method to test the condition of the surfaces of glass plates used in obtaining breath figures, and the following details are repre sentative of the results obtained. A chemically clean watch-glass was placed on an equally clean sheet of plate glass as in Hardy’s experiments, and it was found that a horizontal pull of 8 grams was needed to Dr. T. J. Baker on Breath Figures. 7 start motion. This was not true sliding, but consisted of je: ky movements accompanied by a gritty sound resembling that produced by a diamond when drawn across glass. One half of the glass surface was then rubbed with clean linen and the watch-glass was placed upon it, and it vas found that a pull of 4 grams was sufficient to cause steady sliding without noise. Evidently the linen had contaminated the glass. A blowpipe-flame (coal-gas) was now swept across that half of the sheet which had been rubbed with linen, and once across the unrubbed and therefore chemically clean half. Each of these flame tracks was tested with the watch- glass, and a pull of 5°5 grams was required in each instance to cause movement, and this motion was not smooth. It appears reasonable to suppose that in one case the flame had removed the film of contamination left by the linen, and that in the other it had slightly contaminated the half which was chemically clean. In another experiment the flame of pure carbon monoxide was used instead of a coal-gas flame, in order to eliminate the possibility of contaminating the glass by the products of combustion of coal-gas. Steady sliding of the watch-glass occurred on the linen-rubbed part with a pull of 4 grams ; but when the watch-glass was placed on the track of the CO flame a pull of rather more than 7 grams was required, and the movement was of the type associated with a clean surface. This pull of 7 grams is very close to the value (8 grams) required on chemically clean glass. If a thin clean glass rod with a rounded end is drawn gently across a glass sheet which has been rubbed with clean linen it slides freely and noiselessly, but when it encounters a flame track the increased friction is easily felt, and a faint squeak may be heard. 7 Lc Chemical deposition of silver on a flame track. If a film of silver is deposited chemically on a sheet of “cleaned” glass across which a flame of carbon monoxide has been swept, it is seen that the mirror is whiter and freer from pin-holes on the flamed part than it is elsewhere. This points to the greater cleanliness of the flamed portion. A test-tube was washed out with soapy water followed by tap-water, and was then thoroughly dried by rubbing the inside with clean linen. A narrow belt of the tube near its middle was then heated externally in a small Bunsen flame to a temperature which was much below its softening-point. (ers _ Dr. T. J. Baker on Breath Figures. When the tube was cold a silvering solution was introduced, and the quality of the mirror obtained was better on the zone which had been heated than elsewhere. ‘The borders of this belt were badly silvered, and it seems probable that the con- taminating film which covered the interior of the tube had been driven from the heated belt to the cooler parts on each side of it, where the increased contamination would affect the deposition of silver adversely. If the tube has been heated until softening is imminent, the silver deposits on this part in a manner which suggests that the original smoothness of the glass surface has been partially destroyed. The point which was at issue between Lord Rayleigh and Aitken appears therefore to be decided in favour of the former, who believed that the passage of a flame across the glass cleanses it, and thus favours the condensation of moisture in the form of a continuous film instead of droplets. Nitric acid not a cause of breath figures. It remains to consider the suggestion of Craig, viz. that some nitric acid might be formed in a flame and be deposited on the glass, thus determining the formation of a breath figure. If hydrogen were burnt in pure oxygen this possi- bility would be excluded. A jet of hydrogen was ignited electrically in a large tube through whicha stream of oxygen was passing, and by a simple device a small glass sheet was passed through the flame. The glass showed an excellent breath figure when breathed upon. The oxygen used was prepared in one case from sodium peroxide and water, and in another experiment from potassium permanganate, but absorption with pyrogailol showed that it contained rather more than 1 per cent. of (presumably) nitrogen. The result is not conclusive, because a small amount of nitric acid might have been formed ; but it seems unlikely that the reduction of the nitrogen from 79 per cent. (in air) to about 1 per cent. should have been without influence on the strength of ,the breath figure if nitric acid plays any part in the process ; and it is difficult to see how the presence of a minute quantity of acid could account for the great increase in friction described in the preceding section. Transference of breath figures to a second plate. In that which follows it will be convenient to refer to a plate which has been traversed by a flame as a “flamed ” plate. Dr. T. J. Baker on Breath Figures. 759 A flamed plate which had been used for certain expe- riments was by chance placed with its flamed surface down- wards on another glass plate which had not been used. Next day the plates were separated, and it was observed that both plates gave breath figures, one being an exact copy of the other. Moreover, the original showed no diminution of intensity. This accidental observation was many times confirmed, and it became clear that closer investigation was necessary. The transference of the effect from one plate to another lying upon it suggested that some volatile material was concerned i in the process, and that the escape of this material might be assisted by reduction of pressure or by increase of temperature. A flamed glass was therefore put face to face with a “cleaned ” sheet of glass and the two plates, clipped together, were placed in the receiver of an air-pump, and the pressure was reduced to a few cms. of mercury. A few minutes later the plates were withdrawn, separated, and breathed upon, when a perfect copy of the original was obtained on the “cleaned ” plate, whilst the flamed plate still retained its power of producing a breath figure with unim- paired intensity. Thus transference occurs as effectively in a few minutes under reduced pressure as it would in the course of hours under atmospheric pressure. Hxperiments were then made with plates separated about + mm., and a clear transfer was obtained in about 15 minutes. The separation was increased to about 2 mm., and again a transfer occurred, but much diminished in intensity. With a separation of 1 cm. it was not possible to detect with any certainty that transfer had occurred. The flames of hydrogen and carbon monoxide also pro- duced transferable figures ; and, since neither of these gases yields solid products on burning, it appears unlikely that the volatile material causing the transfer can have been provided by the flame. It was also found that the transferred figure resembles the original in offering marked resistance to the steady sliding of a watch-glass across the glass plate, although the effect, as might be expected, is weaker. Whatever may be the explanation of the phenomenon of transfer, this fact indicates that the transferred figure represents a portion of the glass which has been partially cleared of the contaminating film. At this point it was thought desirable to discover whether the peculiar properties of a flamed plate are modified by breathing upon it. One half of a flame track was covered up 760 Dr. T. J. Baker on Breath Figures. wand the other half was breathed upon. As soon as the deposited moisture had evaporated the protecting cover was removed, and the usual procedure for obtaining a transfer im vacuo was followed. Not the slightest difference in the two halves of the transfer could be detected. It is therefore permissible, and sometimes convenient, to test the flamed plate by breathing upon it before using it to obtain a transfer. Length of time during which a flamed plate retains its power of giving a transfer. A flamed plate was kept in a warm room for 60 hours, and at the end of this time it was left zn vacuo for 24 hours in contact with a “cleaned” plate, and on this a good transfer was obtained. Another flamed plate after 9 days’ exposure to the air behaved similarly, but the transfer was fainter; and ina third instance a plate produced a transfer 18 days after it had been flamed. The loss of the volatile material is plainly very slow under ordinary conditions of temperature and pressure. Condition of the flamed plate after zt has been exposed to reduced pressure. A flamed plate was left in vacuo for 24 hours. It was then removed and clipped face to face with a “ cleaned ”’ plate, and the two were kept in vacuo for 20 hours. No transfer occurred, but the original flamed plate gave a breath figure as good as though it had not been exposed to low pressure. Ina second trial the time allowed for transfer in vacuo was extended to 48 hours, but no trace of transference could be detected. These facts seem to confirm the hypothesis that transference is due to the escape of material from the flamed track on the plate, and that the whole of this escapes under reduced pressure in the course of a day. At the same time it is clear that the surface of the glass which has lost this matter is still in an abnormal condition, and whatever this condition may be it is one which persists for many weeks. Secondary transfer. A transfer was obtained in the usual manner. ‘The plate A on which this transfer had been effected was then clipped in contact with a “cleaned” plate B, and both were placed in vacuo for 10 minutes. At the end of this period they were separated, and plate B was examined. No transfer Dr. T. J. Baker on Breath Figures. 76] could be detected. The plates were again putin contact and left in vacuo for 2 days, and then it was found thata transfer from A to B had occurred. Plate A still retained its power of producing a breath figure. These secondary transfers afford further confirmation of the view that the substance which modifies the glass surface is volatile, and it is worthy of notice that the glass on which the first transfer was obtained still retained its property of yielding a breath figure. Permanent record of a transferred figure. When silver is deposited chemically ona glass sheet which has received a transfer, the pattern is recorded precisely in the same manner as the figure on the original flamed plate. Transfer produced by heating the flamed plate. A flamed plate and a “cleaned” plate were held face to face by clips, and the back of the flamed plate was heated by a Bunsen flame until it was uncomfortably hot to the touch. When cold the ‘cleaned ”’ plate showed an excellent transfer, and the flamed plate itself still gave a perfect breath figure. The original flamed plate was now placed in contact with anothsr ‘‘ cleaned” plate, and the process was repeated. This resulted in a very clear transfer, but rather fainter than the first, indicating that not all the volatile matter had been expelled by the first heating. Experiments were then made with the plates slightly separated, and transfers were obtained even when the distance between them was fully 2 mm., but the outlines of the figures were less distinct, probably owing to diffusion of the volatile matter during its passage across the inter- vening space. Temperature required to expel the volatile matter. The volatile matter is rapidly dissipated at 100° C., for it was found impossible to obtain a transfer froma flamed plate which had been. heated in a steam-oven for # hour. In this connexion Lord Rayleigh’s observation that a breath figure may be obtained on the inside of a test-tube by heating it externally to redness may be recalled. Such a figure should be incapable of transference because the high temperature of the walls of the tube would have expelled any volatile matter which may have been there. To test the point a flat sheet of fused silica was held so that the tip of a small blowpipe-flame impinged on the middle of one face until a 762 Dr. T. J. Baker on Breath Figures. red-hot spot was visible. When quite cold the face of the plate remote from the flamed face was placed in contact with a “cleaned” sheet of glass, and both were subjected to reduced pressure. No transfer was visible even after the lapse of 46 hours. A similar negative result is obtained if the flamed face of the silica plate is used. Hlectric breath figures. - The tracks of electric sparks which have traversed the surface of a “‘ cleaned ” plate of glass are rendered visible by “black” condensation when the plate is breathed upon. Figures so obtained are transferable, and in all their pro- perties resemble those produced by flames. Electric discharges in air produce ozone, but the present Lord ne has shown that this gas does not yield breath figures on glass. Nitric acid is also formed, but the author has found that good figures can be obtained on glass which has been tra- versed by sparks in an atmosphere of hydrogen. It is therefore unlikely that the figures are caused either by ozone or nitric acid. By passinga large number of sparks between two platinum sheets which rested upon a sheet of “cleaned” glass a well-defined area was obtained, which could be tested for friction by Hardy’s method. To cause steady motion on the unsparked portion of the glass a pull of 4] orams on the watch-glass was sufficient, whilst on the sparked area a force of 74 to 8 grams weight was necessary to start motion. These results are almost identical with those obtained on the track left by the flame of carbon monoxide (p. 757), and it is highly probable that sparks remove the contaminating film from the glass surface either by their heating or disruptive effects. It is well known that when a sheet of cleaned glass is placed upon an insulated metal plate which is connected to one pole of an induction-coil, and a coin, connected to the other pole, is laid on the glass, the passage of a discharge for a few seconds will produce the conditions for the developme:t of a breath figure on the glass. The ‘ black ” condensation corresponds to those parts of the coin which are in relief and, | in addition, the tracks of any sparks which may have traversed the glass surface from the edge of the coin will also be rendered evident as wavy lines of “ black ” condensation. It may be assumed that discharges from the under surface of the coin to the glass beneath it occur most abundantl from those parts which are in highest relief, so that the film | | E — ~~ ~~ ee, ee ae Ie eee eh Dr. T. J. Baker on Breath Figures. 763 of contamination is. removed from those parts of the glass more rapidly than from the neighbouring areas, and a breath figure revealing the design of the coin may be developed. If the time during which the discharge takes place is too prolonged, the whole of the film beneath the coin is removed, all details disappear, and only a disk of “ black ” condensation is obtained when the glass is breathed upon. Hlectrical conductivity of breath figures. The author has found that the passage of a flame or of a stream of electric sparks across the surface of ‘‘ cleaned” glass greatly reduces the insulating property. Two ebonite rods were capped with small pads of tin-foil, one of which was earthed, and the other was connected by a wire to a charged electroscope. By pressing the two pads simultaneously on the surface of the glass to be examined any leakage across the intervening portion of the glass is easily detected. Flaine tracks produced by burning coal-gas, hydrogen, and carbon monoxide all show considerable conductivity. Coal-gas appears to be most effective, and this is not un- likely, because sulphurous acid is one of the products of its combustion. If a flamed plate is heated in a steam oven for about - 30 minutes the conductivity of the flame track is found to have been reduced very greatly, and a similar result is obtained after a flamed plate has been left in vacuo for a few hours. ‘Transfers are also found to possess a certain small conductivity. It is a matter of indifference whether the electroscope has been charged positively or negatively. The reduction in conductivity brought about by heating to 100°, or by exposure to reduced pressure, suggests that water derived from the burning of the coal-gas or hydrogen may. account for the effect, but it is not clear why the flame of carbon monoxide or electric sparks should produce con- ductivity. Discussion of results. It is probable that a breath figure produced by a flame or by electric sparks is to be attributed in part to the burning off or volatilization of the thin film of contamination which ‘Is present on a surface which has been rubbed with “‘ clean” linen. The track of the flame or spark then presents an 764 Des, a: Baker on Breath Figures. uncontaminated surface on which moisture condenses in the form of a continuous transparent film. It would be expected that the flame-cleaned track would speedily become contaminated again and cease to function, but the extraordinary persistence of the property associated with the production of a breath figure (p. 756) indicates that other factors have to be considered. What these factors are cannot be asserted with confidence, but it is not improbable that the structure of the surface layer of the glass itself suffers a change during its momentary exposure to a high temperature, and it is also possible that some of the decom- position products of the contaminating film are occluded by the glass along the flame track. If a chemically cleaned sheet of glass is traversed by a flame of carbon monoxide, and, when quite cold, is immersed in a silvering solution, it is found that the silver begins to deposit first along the flame track. Since no film of contamination previously existed on the glass, it would appear that the difference in the rate of deposition of silver is due to a physical change in the surface of the glass. It is more difficult to offer an explanation of the trans- ference of a breath figure from a flamed plate to a ‘‘cleaned” plate, but since the process is hastened by reduction of pres- sure and by rise in temperature, and occurs even when the plates are not in contact, it is clear that some gasecus material passes from one to the other. Also, it has to be borne in | mind that the transferred figure is an area from which the contaminating film has been more or less removed (ped ag)). We may imagine that the contaminating film ona “‘ cleaned” plate tends to prevent the ready escape of gas-molecules which have been occluded by the glass surface, but that where this impediment has been removed by the passage of a flame, or by sparks, a violent outrush occurs when the temperature is raised or the pressure is reduced, and these molecules on striking the opposed surface of the “cleaned ” plate break up and scatter that portion of the contaminating film on which they impinge and thus expose a relatively clean surface on which moisture will condense in the “black” form. This suggestion may be extended to explain a second transfer from the first. The molecules which are active in producing this result probably arise from the occluded products of decomposition of the contaminating film, for it has been found that no transfer, or at most a very faint one, can be obtained from a chemically clean plate which has been flamed. a a ae a a ll Repulsive Eject wpon Poles of Electric Are. 765 The results of the experiments on the electrical conduc- tivity of flame tracks and of their transfers point to water molecules being one of the active substances, but the experi- ments with carbon monoxide and with sparks show that the effects caunot be attributed to water molecules alone. The author has obtained some evidence that the transfer in vacuo of a breath figure to the sensitive surface of a photo- eraphie plate is capable of development, but the necessary conditions are as yet uncertain and require further investi- gation. King Edward’s School, Birmingham, Feb. 21, 1922. LXVI Repulsive Effect upon the Poles ee the Hlectric Arc. By A. SELLERIO * 1. FN December 1919 Prof. W. G. Duffield published the results of a careful series of experiments carried out in conjunction with Messrs. Burnham and Davis, on the same subject as the present paper f. As similar experiments made by me are not mentioned there, it seems that my Notet of 1916 is unknown to the authors. It may be useful to put together both the results concerning this interesting subject§: but before doing so, I must observe that in “evaluating my readings I did not take account of the electromagnetic force V due tv the earth’s magnetic field, and of the electro- dynamic action Ii between the fixed and the movable parts of the circuit. In recommencing my experiments some years ago, I remarked that V, EK were on the contrary not to be neglected, and in the meantime I was notified of Prof. Duffield’ Ss work, in which the different sources of error are accurately separated. His investigation enables me to estimate the corrections concerning my results, without further trials. * Communieated by the Author. + Phil. Trans. Roy. Soc. of London, A, vol. ccxx. p. 109 (1919). A further note on metallic and composite arcs is recorded in Science Abs. 1920 (Roy. Soc. Proc. xevil. p..326 (1920)). - { A. Sellerio, “ Effetto di repulsione nell’ arco elettrico,’ MNwovo Cimento, xi. p. 67 (1916). This paper and the first noticed of Prof. Dutheld will be denoted by the initials 8. and D. § Dutheld’s research was the subject of a note by Mr. Ratner, Phil. Mag. xl. p. 511 (1920), and Prof. Tyndall, zbedem, p. 780. 766 Dr. A. Sellerio on the Repulsive Effect upon It may be well to give an outline of my apparatus : a sort of torsion balance, shown in plan in fig. 1 (from the Nuovo At O were connected to the horizontal rod a suspension fibre normal to the plan hanging down from a graduated Fig. 1. Sehermo se im) -| b ! Ne m te 0 6 : ue u a Contrappeso Microscopio torsion head, and a little iron style dipping into a mercury trough. Therefore the current flowed through the mercury either in the direction OMBA, or in the opposite one. Further details are referred to loc. cit. When the arc is started, the arm OM tends to recoil, and to hold it stationary it is necessary to give to the fibre a certain torsion «, corresponding to a force R= 0°036 « dyne. In the force F there are to be distinguished :— The true repulsion rising within the are gap ; Some disturbance occasioned from the heat, as air convection currents, ete. ; The influence V of the earth’s magnetic field ; The electro-dynamic action E between OMB and the - fixed circuit. 2. The earth’s action V can be easily calculated, as follows :— As the arm OMB is free to turn about a vertical axis, the recorded forces are only the horizontal one, and consequently the earth’s field is acting upon OMB only with its horizontal the Poles of the Electric Are. 767 component, H,. When B is +, the current direction is OMB, thus according with Fleming’s law, the repulsion is apparently increased. The resultant H,.OM.z2 of the forces acting on OM is applied to the centre of OM, hence on transferring it to the end M it is to be reduced to In a similar way, the couple due to the forces acting on BM being (H,. BM .2z) a, we may replace it with a force P_, 7. e. a greater effect upon the anode. 5. In order to find out how other circumstances may influence the pressure P, I have tried some expsriments with cored carbons, finding an increased effect upon the negative pole and a reduced one upon the positive. This behaviour is to be attributed to the metallic salts of the core (S., p. 77). I have also noticed that the readings for ascending and for descending current are often a little different, as happens, for instance, in P. D. measurements, for both the shape and the matter of the carbon (occluding gases, metallic salts, grain, &c.) are altered by burning. The diameter of the electrodes has no great influence in the present research, of course only while it remains large relatively to the crater size. For, putting a carbon rod 12 mm.in diameter against a similar one of 3 mm., when the latter is acting as anode, the arc hums and the repulsion becomes greater (S., fig. 8). Further remarks, made also with thick carbon rods either ~ * In Duffield’s experiments, setting approximately H,=0°47 (Ingland) and taking OM=OB=11 cm. from his hgs. 1 and 17, formula (1) would give V=0'261. Instead of 0:26, w2 get from fig. 8, 0:14, and from p- 124, 0°48. Phil Mag. 8. 6. Vol. 44. No. 262. Oct. 1922. 3D 770 Dr. A. Sellerio on the Repulsive Effect upon by increasing the current strength over the hissing point, or by shortening the length L, &c., have generally shown that when the are is not quite steady and silent, the forces acting upon the poles become greater. 6. In order to test whether the observed effect P is due to any disturbance C produced by heat, as convection currents of hot air, an arc was struck between two fixed carbon rods A, B (fig. 2, from the Nuovo Cim.). No repulsion was exerted upon a movable carbon B’, although it was so near to B' as to become white hot. On the contrary, the influence of the heat is to cause the poles to approach each Fig. 2. M O other ; the deflexion due to C was 10 to 20 degrees, or 0°36 to 0°72 dyne (S., p. 70). Then if we momentarily denote with O the observed effect, the true repulsion is to be set, P=O+C. = This disturbance is not entirely avoided by enclosing the apparatus in a box, to prevent air currents, as stated by Prof. Duffield. He tried to determine the value of C, and, having found approximately 0°44 dyne at 8 amp., proceeded to draw a correction curve C. However, “It is unfortunate that information is very difficult to obtain in the crucial part of the curve, where the current is small,” as he says, thus we cannot yet decide with certainty whether the true repul- sion P is present witn the smallest currents, or whether it starts up only at a current minimum. This interesting question is included in a general one, the Poles of the Electric Arc. foie ae. either the repulsion P is a gradual and uniform effect, or not. Fig. 3 shows a typical specimen of my readings. As the corrections Vee C, whatever their exact values may be, do surely vary in the same way on increasing the Fig. 3. 600|% =f 560 - Pi carbonce+ 12mm ) Negat iyo 720 ° Wied normate 80 x >> atbi lante e } x x Amp. x Bl eae ay Dai Oe Be ei Oh AB RO current, i. é. they as well as their derivatives are both con- tinuous functions of I, the ‘‘ Knees” remarked on graph 3 must remain after allowing for the corrections V, H, ©, and consequently they are to be attributed to the true repulsion P. If the explanation of P as a recoil effect by the departure of carbon ions or molecules is accepted, the different traits of the curves « or P are easily to be explained (S., p. 86) by assuming that when the current exceeds a certain value greater particles are also expelled from the craters. aD2 772 Dr. A. Sellerio on the Repulsive Effect upon 7. The next point to be considered, as I pointed out loc. ct., is the specific pressure p = = that is, the pressure upon the unit surface of the crater, A being the whole area. Taking from Duffield’s fig. 22 the values P (as unaffected by E, V, and corrected for C), and, on the other hand, from my fig. 10 the values A fora circular crater normal to the carbon rod BM, we obtain TABLE II, 1, Pa 4. 6. 8. 10. ann 2U0 DC \ P O10. “044° 090 4-430 Osea Cathode ene eo Z ‘ yne AMOCOIN ue one, 0:017 0:088 0'16 0:24 0°31 em.” Oathode A ............ 0:005 0027 0050 0073 0096 >> ANNO DOD ea Stara 5°83 50 56 59 68 dyne/em.? Cathode p 2. s:-.- 20 16°5 18 19°5 22 >> This table shows that the unit force does not vary much. with the current strength. It must not be forgotten that, owing to the uncertainty in the data P, A, the above values are only recorded in order to give a rough estimate of the specific pressure p. To pursue accurately this inquiry, coherent values of P and A for the same carbons and the same arc length are required. Besides the mechanical pressure, recent manometric obser- vations with drilled carbons have shown for current strength under 20 amp. * a hydrostatic pressure up to 30 dyne/em.?, an order of. magnitude not far away from that of the mechanical pressure p. In a research f on the electric are Hgt/C~ between Hg as anode and a thin carbon rod as eathode, I have calculated a pressure of 6500 dyne/cm.’? upon the positive pole Hg. In fact, it is known that in a mercury are a cavity of 1 mm. or more in depth has been often observed }, corresponding to a pressure range over 1300 dyne/em.’, with currents of a few amperes. With a carbon-carbon arc the specific pressure p is, as shown, a hundredfold smaller. * H. E.G, Beer and A. M. Tyndall, Phil. Mag. xlii. p. 956 (1921). t A. Sellerio, ‘‘Contributo allo studio quantitativo dell’ arco eletrico fra mercurio e carbone,” Nuovo Cim. xxiii. jan —febr. 1922. { Stark u. Reich. Phys. Zeit. iv. p. 823 (1902). See also, Stark u. Cassuto, Phys. Zeit. v. p. 269 (1904). | ie ew oS he the Poles of the Hlectrie Are, aie ON THE NATURE OF P, 8. For the purpose of explaining the observed mechanical pressure, many hypotheses have been suggested *, starting from different points of view on the arc mechanism, and a conscientious discussion would carry us beyond the limits of the present paper. Without any assumption, it may be observed that of course in the arc a loss of matter by each electrode anda transport from anode to cathode occur, thus—on the anode at least—a recoiling effect must be occasioned. Whatever the nature of the forces propelling the carbon particles may be, the recoil due to evaporation can be estimated as follows :— Let N, p, v be respectively the number of particles leaving the crater in 1 second, their mass, and their velocity of pro- pulsion; then the considered recoiling effect is given by Nee ene eo aoa (Al) By pw, v mean values are to be understood, for it is quite improbable that all the particles possess the same mass and are projected with uniform speed, therefore the right-hand side of (4) is a substitute for an expression Ynpv, with N=2n. In calculating the repulsion P as a recoil effect, by a formula similar to (4), Prof. Duffield assumes for v the velocity of agitation of carbon atoms at the temperature 4000° C. of boiling, that is, 2 4273 : 18°39. 10+ Ve \ 973 at 2°97 1G? em. /Sec., 18°39.10* being the molecular velocity of H, at 0°. It seems to me that this assumption is hardly defensible, as will be best shown by means of the following analogy. If we keep a compressed gas in a bulb, by opening a tap the gas escapes, Impressing a reaction upon the bulb, as in tur- bines. The velocity responsible for this recoil is not at all the molecular velocity of agitation (a function of the gas temperature), it is the velocity v with which the gas departs from the bulb (a function of the pressure difference in and out of the vessel). The value v will be obviously less than the molecular speed ; thus the pressure P does not reach to the high range estimated by Duffield. * Besides the works already noticed, see a recent paper of Prof. A. M. Tyndall, ‘On the Forces acting upon the Poles of the Electric Arc,” Phil. Mag. xlii. p. 972 (Dec. 1921). 174 Dr, A. Sellerio on the Repulsive Effect upon In my paper on the are Hgt/C~, above mentioned, assuming for the vapours issuing from the anode the boiling temperature 357° C.=630 abs., I have found a value V =1000 em./sec., whilst at the same temperature the atomic speed would be v2 7 630 . 4 eee ee OR Iksseayg) = Iu cole 973 7 28,000 cem./sec. Even when the assumed temperature at the Hg crater is really higher, both the considered speeds remain too different from each other. 9. With a C/C arc, if the carbon consumption did occur only by evaporation in the crater, we should have, denoting with d the mean vapour density, a loss of mass per second, m = Avd. As, however, in an ordinary are the carbon out of the crater is consumed also by burning in the air, we must write m= Ard, whence m With reference to some of Duffield’s researches”, we can take for the anode with 1=8amp. and L=3mm., m = 135.10-° g./sec., the area A being 0°24 cm.’, as shown in Table II. The density of the carbon vapours at 4000° C., being taken as 0°00009, the abs. density of Hs: ataO nee becomes 12 0-00009 ie 273 then we have from (5) ” = 650 cm./sec., a velocity range far removed from the atomic speed 2°97 . 10° mentioned above. We will calculate in a simple manner what should be the average velocity v, so as to give account of the observed effect P. * ‘Consumption of Carbon in the Electric Arc,” Roy. Soc. Proc. A. vol. xcii. p. 122 (1915). Above data are taken from a “ Note upon the Alternating-Current Carbon Arc,” by Prof. Duffield and Mary D. Walker, Phil. Mag. vol. xl. p. 781 (1920). the Poles of the Electric Arc. 175 | If we momentarily assume as known the net carbon con- sumption m, t.e. the loss occurring by evaporation at the crater surface in 1 second, we can write instead of (5), the equation m = Avd. It follows from (4), with m = Nu, ee een or for unit surface peat eee ce ad f ay Oe Pier p : a= et Tae ede Taking p from Table II., we get a mean value V = 400 cm./sec. in a good agreement with the foregoing upper limit 650. It the observed p is half due to recoil and half to impact of particles moving towards the electrode, v becomes still less, i.e. 280 cm./sec. The values 400 or 280 mean that if 400. 2 650°" ° per cent., or 65 waste occurs in the are gap by evaporation, the calculated pressure agrees with the observed. S i.e. 43 per cent. of the carbon | 10. Hitherto we have made no hypotheses on the electrical nature of the particles issuing from the crater. It may be pointed out that the repulsion P, owing to the intensive evaporation due to heat, may be also produced from neutral carbon particles carrying no current, a trivial phenomenon. If we assume, on the contrary, that the particles possess an electrical charge, and that the arc mechanism may be in the first instance reduced to a stream of positive ions of mean charge e moving from anode to cathode and carrying a current portion, al, and to an inverse stream of negative ious (or electrons) carrying the rest (1—a) .I, a system of equations may be written *, from which it follows € al - = U . . ° ° e ° . . g vies (9) * S., p. 80. To correct 6,W,=iN,4,v,2=0,2,.TE. According to Thomson (‘ Conduction’. ... p. 426 (1903)) the ratios a, (1—a) should be proportional to the mobilities of the positive and negative carriers. 776 Repulsive Effect upon Poles of Electric Arc. As O=axl, we have é vl . ; ! Se Bige (9) and putting v= 400cm./sec., I= 8amp.=0°8 E.M.U., P= 1:43 dyne (Table II.), the ratio e/w for the positive particles issuing from the anode results : = 225 B.M.U. Ue gly a ell? of the cheeened is Ge dp noe, : < 500. The value e/w being 9580 for hydrogen atoms, becomes 9580 oe 740 for carbon atoms if carrying one elementary charge. Then the range 255 (or 500) does not conflict with the values theoretically admissible. i may be observed that the meaning of values e/p less than the theoretical is that carbon particles on starting from the crater are not fully disintegrated and ionized. Holding a different point of view, Prof. Tyndall has also reached the conclusion (loc. cit.) that the observed pressure P can be best accounted for as a recoil) by departure and by impact of carbon ions. The electrons, probably, contribute very little to the mechanical pressure P. SUMMARY. The experiments of Dufheld, Burnham, and Davis with carbon-carbon are are generally in a good agreement with ‘mine. The main results are: 1. In the electric arc there is a repulsive effect upon the poles, increasing with the current. The range of P is less than 10 dyne with currents up to 20 amp. P does not vary much with the arc-length L, except when L=~0, when P becomes evidently greater. 2. The carbon quality hasa great influence on P. It seems that metallic salts cause an increase in the pressure on the cathode and diminish that on the anode (S.). 3. With uncored carbons the repulsion on the cathode appears smaller than that on the anode. On the contrary, the specific pressure p per unit crater surface is greater on Path of an Electron in Neighbourhood of an Atom, 777 the cathode (S.). The range of p, and p_ is about 10 dyne/em.?. 4. It is yet not certain whether the law connecting P with I is a linear one, and whether the pressure does arise with every current strength (D.), or requires a current minimum (S.). Probably with increasing I, the pressure P does not increase by a uniform law, for the graphs show some “' Knees”, which suggest different are stages (S.). 5. The value P calculated, in testing the recoil hypothesis, by taking for the velocity of the carbon particles their atomic speed, is too great (D.). Whilst, by taking the propulsion velocity of carbon atoms starting from the positive crater, the calculated repulsion is in a far better agreement with the experimental results (S.). 6. The propulsive velocity of carbon particles has been estimated as 280-400 em./sec. (8.). In conclusion, I think we cannot yet say with full knowledge whether the observed effect is intimately associated with the electrical processes of the arc, or whether it simply accom- panies in an ordinary way the evaporation of electrodes at high temperature. Only after having established the former view by further investigations, available information on the nature of the are will be given by P measurements, im- proving the theoretical construction whose foundations have been established by Thomson, and Stark. Istituto fisico della R. Universita, Palermo, 22 March, 1922. LXVIL. The Path of an Electron in the Neighbourhood of an Atem. By Bevan B. Baxer, A, BSc, FLRSL., Lecturer in Mathematics in the University of Edinburgh *. i: XPERIMENT has shown that when an electron collides with an atom, thereby causing it to emit radiation, the frequency v of the radiation is related to the amount U of the kinetic energy of the electron absorbed by the atom by the equation Why. where / denotes Planck’s constant of Action. Professor Whittaker f has recently shown that, in order that all * Communicated by the Author. + E. T. Whittaker, “On the Quantum Mechanism in the Atom,” Proc, Roy. Soc. Edin. xli. pp. 180-142 (1922). 778 Mr. B. B. Baker on the Path o7 an exchanges between the kinetic energy of the electron and the radiant energy should conform to this relation, it is necessary that the atom should contain a mechanism which is such that an electron approaching an atom will induce in the atom what may be called a ‘magnetic current” ; the model which he has suggested to typify such a structure consists of a number of elementary bar magnets lying in a plane and rigidly connected like the spokes of a wheel, so that they rotate together in the plane, each magnet having one pole at the centre of the wheel and having its direction at every instant radial from the centre ; it is, in fact, such a structure as Sir Alfred Ewing has proposed to explain induced magnetism *. If we suppose such a magnetic wheel to be placed with its centre at the origin and its plane in the plane of yz, and - suppose an electron to be projected towards it along the axis. of w, then the electron, by its motion, creates a magnetic field which will cause the magnetic wheel to rotate, and the rota- tion of the magnetic wheel will set up an electric field which will retard the motion of the electron. Denoting the radius of the magnetic wheel by a, the magnetic moment of one of the elementary bar magnets by wa, the sum of the values of pw for all the elementary bar magnets composing the wheel by M, the moment of inertia of the wheel about its axis by A, the charge on the electron and the mass of the electron by e and m respectively, then Whittaker has shown that if ae 2¢ velocity of projection wt) of the electron is less than ——= m the collision between the electron and the wheel is in the nature of an elastic impact, 7. ¢., the electron is stopped at a certain point and forced to return along its path, the mag- netic structure giving back to the electron the energy it had : , 2eM previously received from it; but that if a> a m electron 1s able to pass completely through and away from the magnetic structure so as to be free from its influence, and the magnetic structure is left in rotation. In this latter case, the amount of energy U lost by the electron and gained by the wheel, is given by the equation 2e?M? U= uae | and the absurbed energy appears in the atom as a magnetic * Cf. Hwing, “On Models of Ferromagnetic Induction,” Proc. Roy. Soc. Edin. xlu. pp. 97-128 (4922). Electronin the Neighbourhood of an Atom. TT9 ; 2eM current, specified by the angular velocity O= a , so that the absorbed energy and the angular velocity are connected by the equation ry U=eMO. Whittaker has further shown that the disturbance in the atom after the collision consists in the displacement of a single electron, and that the radiation emitted by the elec- tron in its oscillatory subsidence to its normal state must satisfy the equation U=hv. 2. In his paper Whittaker has assumed that the electron is projected towards the magnetic wheel * in a line per- pendicular to its plane and directly towards its centre. If we suppose the atoms to contain such structures as have been described, we must suppose the magnetic wheels in the substance to be bombarded to have ail possible orientations, and the electrons to be projected from any direction. It is therefore of interest and importance to discuss the general ease, when the electron is projected in any direction and passes in the neighbourhood of one of these magnetic wheels. We will suppose, as before, that the magnetic wheel has a radius a, and that if wa is the magnetic moment of one of the elementary magnets, the sum of the quantities w for all the magnets forming the wheelis M. Further suppose hat the plane of the wheel is the plane of yz and that the wheel is free to rotate about its axis, which is the axis of a; the wheel is therefore restricted to have only one degree of freedom. Let the amount of rotation of the wheel at any particular instant be specified by the angle y between the axis of y and a definite fixed radius in the plane of the wheel, the angle being considered positive when it is such as would turn the axis of y towards the axis of z, the rectangular axes of wyz torming a right-handed system. Let the moment of inertia of the wheel about its axis be A, so that when the wheel is rotating with angular velocity w its kinetic energy is $Ay”. Let the mass of the electron be m, its charge e, and let its position at any instant be specified by spherical polar coordinates (7, 0, 6) connected with the rectangular * Note.—Whioen referring here or elsewhere to a magnetic wheel, it is to be understood that it is not suggested that an atom actually contains a mechanism similar to that here described, but merely that the atom behaves as if it contained such a structure. 780 Mr. B. B. Baker on the Path of an coordinates (x, y, 2) by the relations x=reos6, y=rsin@cosd, z=rsinOsin db; the kinetic energy of the moving electron is therefore lin(a? + 1°62 + 7? sin? 6°). te We have further to determine the potential energy of the system due to the mutual interaction between the electron and the magnetic wheel. To do this the magnetic wheel, when it is rotating with angular velocity ab, may be looked May upon as a magnetic current of strength —5 flowing i in a circle of radius a. Now just as an teenie eee flowing round a circuit may be replaced by an equivalent magnetic shell bounded by the circuit, whose magnetic moment per unit area is proportional to the current-strength, so we may replace the magnetic current by an electric shell, bounded by the circuit, such an electric shell being equivalent to a charged condenser in electrostatics. For convenience we shall. suppose the electric shell to have the form of a hemi- — sphere of radius a bounded by the circumference of the magnetic wheel, the charge per unit area on either plate of the condenser being — The electric potential at any point P due to the condenser is therefore Be .@, where w is the solid angle subtended by the magnetic wheel at the point P. The potential energy V of the system is thus Mee ; it is independent of the coordinate ¢, and may be eae in powers of 7 in the form: when ra, ays? VeMype [55 P, (cos) — 5-5 & 7 P3(cos 8) +.. 2 1\ de oe In— oe aa eee ee ee 1 (cos é)+...|. Electron in the Neighbourhood of an Atom. 781 We shall write V=MweF (r, 0), where F(y, @) is a function of x and @ alone, and F(*, (= =, where @ is the solid angle subtended by the magnetic wheel at the point (7, 6) The Lagrangian function L=T—V, where T and V are the kinetic and potential energies of the system respectively, is given by the equation L=4AwW? + 4in(r? + 7°? +7? sin? O62) — Mew F(r, 0). (3) The equations of motion of the system are therefore . _ue(2F ) Aah —Me(S. ; +, CONES te tac?) oe Ne (4) m(7*—r6? —r sin? Of?) + Moyo" a. | ) 2 4 o=- “e030 -+ 5 9; (5cos*—3 08 8) — ..., | OL ey 1a 3 : a é ere Zsind+* 4 (15 cos @— 3) sin 0 “a =0( only when cos0=0, 7. ¢., oF Or and therefore, when ry when 0= = or 0= 2 and oe =0 only when sin 9=0, 7. e., 2 when 0=0 or 0=7. In general, therefore, equation (19) will only be eto. when 7=0 and 6=0 simultaneously, 7. e., when all the kinetic energy of the electron has been given up to the wheel. 00 Electron in the Neighbourhood of an Atom. 785 There may, however, be certain exceptional values of », 0, 7, 8 which will satisfy equation (19), even though the electron continues in motion ; that even in these circumstances the electron will pass through the magnetic structure on its return path may be demonstrated thus. Denote by a, and a the accelerations of the electron in the directions of + increasing and 6 increasing respectively ; then equations (11) and (12) may be written Me ; oF 4,= ee je ° e ° . (21) Me - oF w= — ty 8. ° ° . . . (22) Suppose, for definiteness, that the electron is projected from a part of the plane of wy for which both wand y are positive ; then w will be always positive and from equations (20), (21), and (22) we obtain the results: whenr>aand O0<6@< = aa 0 amd: ap. 0; when r > a and ager a Eee Oranoseg > .U: when r>aand w O-and a’ < Os: when r >a and wheny O and ap< 0); ite whens Gand - ore ms p< 0 ardso, < 0; whenr0; OT ; when r <@0<273; a> QO and # > 0. a The radial acceleration is therefore directed away from the centre of the magnetic wheel when « is positive, and towards the centre when w is negative, and thus always tends to retard the motion of the electron on its outward journey. Moreover, when z is positive and r>a the curvature of the path is towards the axis of y ; when « is positive and ra the curvature of the path is towards the axis of a. Now suppose that the electron has penetrated the mag- netic structure and reached a point. for which 7>a (if it only reaches a point for which ra its motion. will be such as to bring it to an even more favourable position for passing through the magnetic structure on the return journey. 6. It has therefore been shown that the electron can permanently transfer an amount of energy to the magnetic wheel only if its velocity and direction of projection are — such that it penetrates the magnetic structure and passes away out of the influence of the wheel without returning on its path. In such a case the wheel is finally left in rotation ; 2M with an angular velocity Q= = and the amount of kinetic energy U transferred from the electron to the struc- Ze ; ture is given by Ue in order to attain this result the initial velocity of the electron must be at least as great 2Me / Am These results are precisely the same as those obtained by Whittaker in the particular case which he considered, and his further discussion of the way in which this absorbed energy is converted by the atom into radiant energy and the deduction of Planck’s relation connecting the energy and the frequency of the emitted radiation may equally be applied in this more general case, as r 737 | LXVIII. On the Theory of Freezing Miatures. By ALFRED W. Porter, D.Sc, F.AS., Funst.P., and REGINALD E. Gipes, 6:Sc., A.lnst-P* 7s 1874, Professor Guthrie carried out extensive experi- ments on freezing mixtures, and was the first to point out the fallacy of a belief which has persisted till even the present day. Guthrie said:—“In regard to freezing mixtures, I confess to have been here very much misled by the confident but rather erroneous statements of others, to which I attached faith trebly blind,—blind, because no re- corded experiments really support them, blinder still because a little thought in the right direction must have shown their fallacy, and blindest of all because the one experiment of my own in this direction shows that the minimum temperature of an ice-salt eryogen is reached, whether we take the ratios three of salt to one of ice, or one of salt to two of ice, and so points to the wideness of the margin of ratios which may obtain between the weights of ice and the salt” f. He showed that the same temperature, viz. the cryohydriec, was reached for a wide range of proportions of the constitu- ents, and that the initial temperature of the salt need not be zero; in fact, in his extreme case the salt was initially at a red heat. The present object is to consider an equation representing the heat changes which occur in a freezing mixture, and to illustrate how well it bears out the truth of Guthrie’s remarks. For the sake of simplicity it is best to concentrate one’s attention on a definite mixture, say that of salt and ice, and to suppose that initially the constituents are all at 0° C. and present in the following amounts: ice I gm., water—>zero, salt S gm. The vanishingly small quantity of water is introduced only to ensure that there will not be any discon- tinuity under the conditions which accompany the reaction. All possible cases can now be divided into three sections :--- (1) That in which the masses of ice and salt are such that there are both free salt and free ice present at the end of the change of temperature. (2) That in which there is no salt remaining. (3) That in which there is no ice remaining. * Communicated by the Authors. + Proc. Phys. Soc. of London, vol. i. Jan. 18, 1875, 788 Prof. A. W. Porter and Mr. R. E. Gibbs on the Case (1). At the end of the change of temperature, let the mass of solution be (M+m) om., hers ie of water, m= mass of salt in solution. Since the solution is in equilibrium with the salt, it will be saturated, and as it is in equilibrium with ice, it will be atthe freezing-point ; hence the final temperature must be the eryohydric. | | | In determining the connexion between M and m at the end of the change, the external work done can be neglected owing to the very small change of volume at the moderate _ pressure of one atmosphere obtaining during the experiment. _ In these circumstances the heat change in a cycle can be taken as zero with sufficient approximation ; ; or, in other words, the particular path of transformation is immaterial so far as heat changes are concerned. Representing the cryohydric temperature by —r° C., the heat equation will be 7(1s;+Sss) =ML,+mL,, where s; and s, are the specific heats of the ice and salt respectively, and where L, is the latent heat of fusion of ice at —r° C., and L, is the latent heat of solution of salt at —7° C. In writing this equation, all the salt and ice has been assumed to cool down initially to - 7° C. and the transformation to take place then at this low temperature. Nothing would be gained by aiming at meticulous accuracy in regard to numerical values. The general trend of results can be illustrated by using constant and approximate values for which the calculations can be made easily. As the solubility of salt varies very little with temperature, one can assume m/M to have a constant value 4. Assuming also the following approximate values, pe LI Oe ee Li 00) L.=6 fat “35° C.) ae boas one obtains Maan 5 ae } Olw * In calculating Ly at —r the formula = 1—s has been em- or poset It would even be erroneous to employ the more usual equation a - 7 =l- 8, because this gives strictly the latent heat under equilibrium conditions, 2. ¢. under a pressure corresponding to a melting- point of —7, whereas the pressure is approximately atmospheric throughout. Theory of Freezing Miatures. 789 | Q Hence, if S is small, a ean be substituted for = whence ma oe ks : and therefore Me eee 5 This result is applicable, provided S¢ — On the other hand, if I is small compared with 8, one obtains epee Paths) mt: and M= = ae These results indicate that the cryohydric temperature will be attained, provided (i.) S€ay 1 +8) and. an) fl +5). This is in agreement with ordinary experience, though the fact that they do not both totally disappear is usually attributed to the substances not having been taken in the Fig. 1. Oy 40)" 26. SOaee BO \Go ap. = 80. 9p... 100 Percenk Salt &—O Theoretical, P26 oe rich pd bake Sate Exper/me nfal, proper proportions. The above work, however, shows that no such “proper” proportion exists; and, in fact, the experimental results are of the kind one would expect according to theory. In fig. 1 this range is represented by the central horizontal portion of the curve. 790 Prof. A. W. Porter and Mr. R. B. Gibbs on the Case (2). No salt remaining. As shown in the above equations, this means that in the original mixture, S is less than 34 of the whole. ‘The final temperature must be the freezing-point of the solution, as ice is in equilibrium with it, but it will not be the oryohy rdric temperature, as the solution is unsaturated (except in the limiting case). The * general equation can be adapted to this case by writing m=S ; 7(Is;+S8s,) = ML, +81. The law connecting the fr eezing- -point with the concentration, 63S M+S_ The application of this law gives 63S Was (Isi+Ss,)=ML, + SI. This has first to be solved for M, and then 7 calculated from the previous equation. The values of 7 for various values of IT and S are shown in the following table :— at least for dilute solutions, is T= TABLE I. 8. I. M. Te 1 gm. 99 gm. 6°12 gm. 875° C. us wey Sy, 12 ae. 3, Ohiaiss 10:0); EG ony 4. ome oe 16°5 In fig. 1 this range is represented by the left-hand sloping portion. of the curve. Case (3). No ice remaining. This necessitates that the mass of ice taken is less than #5 of the whole mixture. The final solution will be saturated because it is in equilibrium with the excess salt. On the other hand, it will not be at the freezing-point (except in the limiting case), as it is not in contact with ice. Again, the general equation can be adapted this time by writing M=I. Thus t(Isi+ Ss,) =ILy+mLs. The final concentration is of course m/I, and will be approxi- mately one-third. Hence Ss Ae + 5) =1(70 12 Ble or approximately, as [ is small, _ 3601 i [2s : Theory of Freezing Mixtures. 791 In fig. | this range is represented by the right-hand sloping portion of the curve, which is practically a straight line. Series of experiments were carried out to test the validity of the above work. Not very much importance was attached to the absolute value of the eryohydric tem- perature reached all along the central portion of the curve, as this depended very largely on the purity of the materials used. One set of results is recorded below :— Weight of Salt. Weight of Ice. ae oe =m. em. Salt Tas: Ts ‘80 27°6 2°82 8:0 98 32°2 2°95 12°5 1:20 344 3°37 15:0 LOr7 511 17°3 19°8 5:0 23°0 17'8 20°0 12°5 54:2 18°7 19:8 13-7 50°7 De Poa 19:8 15°6 44:3 26°0 20°8 20-0 45:0 30°8 20°5 21:2 47°0 31:2 20°5 28-0 59°1 32°2 198 36°0 363 49°8 ys . 3882 17:2 68°9 19°9 45°8 4:1 91°8 12-0 46:1 3°1 93°8 70 377 2°3 94-0 30 The above results were obtained by mixing the ice and salt in a small vacuum flask, great care being taken to make the mixing as complete as possible : this was fairly easy until the percentage of salt was high, say over 75 per cent. ; but for high percentages it was probably imperfect at best. The results agree with those expected from theory except for high concentrations of salt ; and, even for such, they are sufficiently close to substantiate the previous work. Poor mixing and the thermal capacity of the vessel would cause such a deviation. In the practical case of the use of refrigerating mixtures, the body to be cooled is always to be taken into account, and it may produce considerable modification. If its thermal capacity is 0, the equation would now read 3 ° Hi 2 +6 | =ML,,+mL;. Ai esd J 792 On the Theory of Freezing Mixtures. It is most convenient to write fiat ye A few curves (tig. 2) have been drawn for different values of O 10 20 30 40 50 60 7FO 80 Go /00, Ferce nfage of Salt, K to show the effect produced. Possibly the most interesting case is to find the value of K for which the cryohydric is just reached. Assuming thatall the salt and ice is used and that m/M=34, we have 21°6 E ab : +0 | =—I16m or G=8:35 =2'1 (13s), and hence iG alle: therefore to cool a body whose thermal capacity is @ down to —21:6° C., the quantities of materials to be used are 0/5°4 gm. of salt and 6/2'8 gm. of ice. In practice it is always necessary to take somewhat larger quantities to allow for the formation of dew on the exposed surfaces. This is a fairly serious factor, as 1 gm. of dew is equivalent in its heat change to roughly 8 gm. of ice. It will be seen, therefore, that if K<2:1, there exists a central horizontal portion of the curve, whilst if K > 2-1, the two sloping lines intersect at a vertex lying on the 25 per cent. ordinate, THE LONDON, EDINBURGH, ann DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. ~ }- ™S \ uxt SERIES. ] fp rT t nT Oe o VEMBER 1922. LXIX. The Emission of Electrons by X-Rays. By G. Saearer, VM.A., 1851 Exhibition Scholar, Emmanuel College, Cambridge * HE general object of this investigation was to obtain more definite information as to the emission of electrons from matter under the influence of X-ray bom- bardinent. (1) Aistorical.—Very shortly after the discovery of X-rays it was shown that all forms of matter emitted electrons when bombarded by X-rays. Later work revealed the fact that, in this emission, the electrons had all velocities up to a certain maximum ; this maximum yelocity can be determined by the quantum equations— tmv? = Ve = hy, ' where e, m, v represent the charge, mass, and maximum velocity of the electrons, v the frequency of the X-rays, V the applied potential, and h Planck’s constant. In addition to this general electron emission, there is a special emission associated with the characteristic X-radiation of the matter bombarded. Until recently there has been some doubt as to the exact nature of this special emission. Experiments by Barkla and the author f failed to reveal any special distribution of velocity associated with these electrons. * Communicated by Prof. Sir E Rutherford, F.R.S. + Barkla and Shearer, Phil. Mag. xxx. p. 746 (1915). Phil. Mag. Ser. 6. Vol. 44. No. 263. Nov. 1922. 3 EF 794 Mr. G. Shearer on the On the other hand, Robinson and Rawlinson * by the mag- netic spectrum method showed that there were present groups of electrons with special velocities. Kang Fuh Hu + also obtained some evidence of the existence of these groups. Simons { carried out experiments on the same lines as those of Barkla and Shearer, and concluded that sub-groups of electrons were present when the characteristic X-radiations of the bombarded matter were excited. Recently the question has been definitely settled by the experiments of De Broglie 9. By the use of the magnetic spectrum method he has shown that the energies of these groups correspond to h (v—p,), h (v—vy,), &c., where v, vg, vy, &e., represent the frequencies of the incident radiation and of the K, L, &c. radiations of the matter bombarded. Similar results have recently been obtained by Whiddington |. While these experiments show that the electronic radiation consists of a general emission corresponding to the “ white ” radiation from the tube and groups of electrons whose energies obey simple quantum relations, they tell us nothing of the magnitude of the emission, nor of how this magnitude depends on the type of matter from which the electrons are ejected. : Laub {J showed that the efficiency of an element asa source of electrons increased with its atomic weight, but made no attempt to obtain a law governing the variation. Moore**, as a result of some experiments on the relative ionizations produced in various gases, deduced the law that the number of electrons emitted per atom is proportional to the fourth power of the atomic number. These experiments were of a somewhat indirect nature, and the range of atomic number investigated was small. The experiments described here were undertaken, primarily, to throw light, if possible, on this question. (2) EHeperimental Arrangements.—In order to avoid the difficulties of interpretation introduced by the use of ioniza- tion methods, it was decided to measure directly the number of electrons emitted by observing the rate at which an insulated radiator acquired a positive charge under the influence of X-ray bombardment. * Robinson and Rawlinson, Phil. Mag. xxviii. p. 277 (1914). + Kang Fuh Hu, Phys. Rev. xi. p. 505 (1918). t Simons, Phil. Mag. xli. p. 120 (1921). § De Broglie, Journ. de Phys. (6) 11. p. 265 (1921). || Whiddington, Phil. Mag. June 1922. q Laub, Ann. der Phys. (4) xxviii. p. 782 (1908). ** Moore, Proc, Roy. Soc. A. xix. p. 887 (1915). _ j oe ean sll mission of Electrons by X-Rays. 795 Fig. 1 shows the final form of the apparatus used. Rays from an X-ray tube enclosed in a thick lead box passed through a small aperture in the box and entered the cylindrical brass examination vessel A through a hemi- spherical glass window W. Inside the vessel, in the path of the rays, was suspended an insulated brass cube C, which served to carry the materials to be examined. In order that more than one substance might be investigated without dis- mantling the apparatus, the cube was made capable of rotation about a vertical axis by means of a ground-glass joint G, which carried the whole of the insulated system. By turning this joint through 90° at a time different materials could be exposed to the action of the rays. The beam of X-rays was such that the cross-section was never as large as the area of the material under examination, so that only the window and the matter on the face of the cube were exposed to direct X-ray bombardment. An electromagnet M was arranged near the window so that any electrons from the window might be bent back into the walls of the vessel and thus prevented from reaching the cube. ‘The inside of the vessel was lined with filter paper to reduce the eftect of the scattered and characteristic radiations from the cube. Inside the vessel and insulated from it was placed a wide- meshed wire cylinder which could be charged to any desired potential. ae 2 796 Mr. G. Shearer on the The vessel was connected by wide-bore glass tubing toa Gaede mercury pump. A charcoal-filled tube was attached and cooled with liquid air. By these means the vessel was kept at a very low pressure so as to make all ionization effects negligible. As an additional precaution, hydrogen was used as residual gas in some of the experiments. he substances to be examined were mounted on the faces of the cube, and were all of sufficient thickness to give the maximum electron emission. The source of X-rays was a Coolidge tube Hi a tungsten anticathode. This was actuated by a Butt induction coil and mercury interrupter. Owing to the smallness of the effect to be measured, it was not possible to use a mono- chromatic source of X-rays. ‘The rod carrying the cube was connected through earthed shielding tubes to a string electrometer. The rate at which this acquired a positive charge was taken as a measure of the electron emission, and was determined for various sub-> stances. In order to correct for small variations in the intensity of the rays during a set of observations, a standard- izing ionization chamber was fitted. This was connected to a Dolezalek electrometer of low sensitivity, a steady deflexion method being used. (3) General Results —The early results with this apparatus showed that the electron emission was of a more complicated nature than had been anticipated. If the cube was allowed to charge up to a considerable potential, it was found that the rate of charging up fell off rapidly at first and only became steady after a potential of from ten to twenty volts had been reached.. Such an effect might have been due to ionization effects, but a simple calculation showed that the magnitude of the effect was very much larger than that due to the ionization of a gas ata pressure of ‘01 mm., and the pressure in the vessel was certainly less than ‘00L mm. It appeared, therefore, that this effect was due to the presence of a large number of slow electrons. When the cube attained a voltage of from ten to twenty volts the electric field was of sufficient strength to prevent the escape of such electrons. The existence of these low-speed electrons was also suggested by certain observations on the effect of the magnetie field used to deflect the electrons from the window. Tt was found that this field reduced the emission from the cube although the stray field at the surface of the cube was not more than a few Gauss. Emission of Electrons by X-Rays. 797 In these experiments the surface of the cube was perpen- dicular to the direction of propagation of the X-rays, and therefore parallel to the direction of the electric vector in the X-ray beam. It was possible, therefore, that there was a large number of electrons whose initial direction was nearly parallel to the surface of the cube. Such electrons, even if their velocities were considerable, might be bent back into the cube by electric or magnetic fields of relatively small strength. This hypothesis was tested by comparing the reducing effect of electric and magnetic fields.when the angle of incidence was 90° and 45°. It was found that the percentage reduction was the same in the two cases. Had the effect been due to the bending back of electrons emitted in directions approximately parallel to the face of the cube, the reduction of the emission would have been larger in the first case than in the second. The conclusion reached was that, in addition to the high- speed electron emission, there exists also an emission of electrons of low speed. After considerations such as these had led the writer to this conclusion, it was found that similar effects had previously been observed by Campbell * in his work on delta rays. In what follows, first the properties of the high-speed electrons and then those of the slow electrons will be discussed. A. [High-Speed Emission. (4) Relative Hlectron Emission from Metals.—The effect of the low-speed electrons was eliminated by charging the inner wire cylinder to a voltage sufficient to prevent their escape from the surface of the cube. The residual effect was then that due to the high-speed electrons. The metals investigated were Aluminium, Iron, Nickel, Copper, Silver, Tin, Gold, Lead, and Bismuth. The X-ray tube was operated under varying conditions, and_it was found that the relative values for the various metals depended very little on the conditions of the tube. This point. will be discussed more fully later. Table I. shows the results obtained, the value for the tin being taken as 100. TABLE I. Rie ese sok, ode tin Petes Mine Oiurune, on, Au. Pb, Biz Electron Emission ......... bie 40 58 2567)" 94° 100 184° 189 Tot Atomic Number ............ ior be ae =e 4G D0" 79, «682 SS Atomic Weight. “2222-002. i Oo oy ar 108, 119 197, 207) 208 * N. R. Campbell, Phil. Mag. xxiv. p. 783 (1912). 798 Mr. G. Shearer on the It is clear that the efficiency of a metal as a source of electrons increases with its atomic weight or number. Fig. 2 shows the results graphically. The relation between the number of electrons escaping from a metal and its atomic number is very nearly a linear one. The electron emission may be expressed with considerable accuracy by an equation of the form n' = k(N—a), where n/ is the number of electrons escaping from the metal, N is the atomic number, and & and a are constants, the value of a being approximately 10. SO ae 50 y | ce Ge @) 20 4.0) 60 50 -Qilida Atomic Numeer. PLeEcTRON EMISSION Since N denotes the number of electrons in the atom, it might appear that this result implies that the number of electrons emitted from an atom is proportional to the number of electrons in the atom with a small correction due to the presence of the term a in the equation. Such an interpretation is, however, not permissible, as what has been measured in these experiments is the number of electrons which succeed in escaping, and not the number liberated from the atoms under bombardment. In the present state of knowledge of the laws governing the passage of electrons through matter, it is not possible accurately to deduce the number of electrons liberated from an atom from the observed number actually escaping from the surface. An approximation may, however, be obtained by ee certain assumptions which are prebably near the truth. | _ Emission of Electrons by X-Rays. 799 In the first place, let it be assumed that the number of electrons escaping from the radiator falls off exponentially with the depth from which they come. If Ip is the initial intensity of the X-rays, 8 the area bombarded, n the number of electrons liberated per unit volume per unit time per unit intensity, and jy, and py the absorption coefficients of the X-rays and electrons in the radiator, then the number of electrons from a layer dv at a depth a which actually escapes 1s dn'=n.8.1,e “t" "de. The total number escaping is therefore n! = nSIo/(m + pe). Hence, since «4, is small compared with ps, pouNel; fey nr If, on the other hand, it is assumed that the number escaping falls off exponentially with the distance traversed by the electrons in the radiator, the following expression is obtained : n' = nSI,/4py * where zy is the exponential coefficient of absorption for the electrons. Whichever of these two absorption laws is taken, it follows that the number of electrons liberated per unit volume is proportional to mw. times the number actually escaping. From this the number liberated per atom can easily be derived. If A is the atomic weight of the radiator, m the mass of the hydrogen atum, and p the density, the number of atoms * The actual expression for 7’ is 5 (uy +12) ene ' ‘ = x = x of i}: 0 w hich, on evaluation, gives n'=inSI, [= pe ats Be log (1+) J. If this is expanded in terms of ee yand 2 is neglected, it gives the result quoted above. pe” p28 800 Mr. G. Shearer on the per unit volume is p/Am. Hence the number of electrons liberated per atom per unit time per unit intensity is nAm >] p e e . i or, substituting for n in terms of n’, k. Amn’ ; p where & is a constant depending on the exact form of the absorption law chosen. If itis assumed that Lenard’s law that “ is constant hoids p under the conditions obtaining in these experiments, the result of these calculations is that the number of the electrons liberated per atom per unit time per unit intensity is propor- tional to the product of the number of electrons actually escaping and the atomic weight of the substance from which they are liberated. It has been shown that the experimental results led to the conclusion that n’ was proportional to (N—10). Hence the number of electrons liberated per atom is proportional to | A.(N—10). In deducing this result several assumptions have been made, some of which are only rough approximations to the truth. Probably the most serious one is that Lenard’s law—that p/p is constant—can be applied to this case. Hven under the conditions of Lenard’s experiments, the result was only an approximate one. A strong argument in favour of its application to these experiments is that it causes the density to disappear from the final correction to be applied to the observed electron emissions. The values for the electron emissions from the elements gold, lead, and bismuth were found to be approximately equal. On the other hand, the densities of these elements are 19°32, 11°37, and 9°80 respectively. If the correction to be applied to deduce the electron emission per atom were a function of the density, all regularity would disappear from the results. If the correction depends only on the atomic weight or atomic number, no such difficulty presents itself. When this result is compared with the only other result so far obtained, a serious disagreement presents itself. Moore * found that the electron emission per atom was proportional * Toe. cit. Emission of Electrons by X-Rays. 801 to the fourth power of the atomic number. This result was obtained indirectly from observations on the relative ioniza- tions of different gases. The heaviest element used was chlorine, so that the range of Moore’s experiments falls almost entirely outside that of these experiments. Had it not been for the fact that Moore’s law fits in well with the absorption law of Bragy and Peirce, we might have con- eluded that both his law and that deduced here are both approximations to a more general law, the former being an approximation holding for elements of low atomic weight, while the latter is a better approximation for the heavier elements. Bragg and Peirce * have shown that the X-ray energy absorbed per atom is proportional to the fourth power of the atomic number of the absorbing element. The com- bination of this result with that of Moore suggests that the number of electrons emitted is proportional to the X-ray energy absorbed, a result pointed out by Moore. The range of elements used in the experiments of Bragg and Peirce was well within that of these experiments although outside that of Moore’s. On the other hand, if the above results are true, no such simple law appears to hold. Even when allowance is made for the approximate nature of the various assumptions made in deducing the final result, it is extremely difficult to see any way in which these experiments could possibly be reconciled with a fourth-power law. In connexion with this result, it is of interest to refer to a result obtained by Kaye+ on the relative efficiency of various metals as anticathodes in an X-ray tube. Kaye found that the X-ray output of a tube increased linearly with the atomic weight of the metal used as anticathode in the tube. Later experiments by Duane and Shimizu t showed that the proportionality was to the atomic number rather than the atomic weight. In these experiments we are dealing with the transformation of electron energy into X-ray energy, while the problem that is the subject of this paper is the inverse one—that of the transformation of X-ray. energy into electronic energy. It has been shown that there is a linear relation connecting the number of electrons escaping from a metal bombarded by X-rays and the atomic number of the metal. These results imply that, when the number of electrons entering a metal is kept constant—as in Kaye’s experiments,—the ee “ay energy emitted is a linear function * Bragg and Peirce, Phil. Mae. xxviii. p. 626 (1914). t Kaye, Phil. Trans. A. 209, p. 128 (1908). ¢ Duane & Shimizu, Phys. Rev. xiv. p. 525 (1919). 802 Mr. G. Shearer on the of the atomic number, while, if the X-ray energy falling on a metal is kept constant, the number of electrons emerging from the metal is again a linear function of the atomic number of the metal. (5) Haperiments with Substances other than Metals.— Certain experiments were carried out on the electron emission from salts. Only a few salts were tried, and the dataare not sufficiently extensive to warrant the deduction of definite conclusions. The chief salts investigated were As,O;, KI, and RbI. These were chosen in the hope that they would throw light on the electron emission from the elements Arsenic, Rubidium, and Iodine, and thus serve to fil! up some of the gaps in the electron emission-atomic number curve obtained from the investigation of the metals. The values found for these salts, with tin taken as 100, were 47, 128, and 158 respectively. The values for KI and RbI were both much higher than is to be expected if the effect is an additive one and the values of the electron emission were such as would result from the interpolation from fig. 2. The atomic numbers of K, Rb, and I are 19, 37, and 53, so that, if the effects are additive, it is to be expected that the values for these iodides would be less than 100, whereas it was found that they were considerably in excess of this value. These salts were placed on the faces of the cube in the form of a layer of small crystals, and it is possible that the effective area under bombardment was thus considerably greater than in the case of a sheet of metal. In spite of this possibility, it seems difficult to account for the high values obtaived for these salts. It is interesting to note that the elements potassium and rubidium are both very active from a photoelectric and thermionic point of view. A few experiments were made on other salts of potassium. It was shown that the electron emission increased with the molecular weight of the salt. Thus, both the carbonate and the sulphate gave very much smaller value than the iodide, while the eftect from the sulphate was greater than that from the carbonate. (6) Special Electron Emission.—It has long been known that when a characteristic radiation of an element is excited there is an increase in the electron emission. Although, in these experiments, no attempt was made to use monochromatic radiations, still it was expected that it would be easy to detect this special electron emission. In order to test for its effect, observations on two metals— especially copper and tin—were made under widely varying r-.- —_—,.):C Emission of Electrons by X-Rays. 803 conditions of the X-ray tube. The voltage of the tube was varied and the ratio of the electron emissions from the two metals was measured. Results of such measurements are given in Table II. The observations have been taken in groups, each group corresponding to a certain range of potential on the tube, the potential being measured by the equivalent spark-gap between points. Tasue II. Range of Spark-Gap. Cu/Sn, Q= 5 cm, 0°57 a= 9 0°55 9-12 _,, 0°55 12> 12), 0°56 It is clear from this table that, under the experimental conditions, the ratio of the electron emission from these elements is practically independent of the potential used to excite the tube. It was expected that there would be a dis- continuity in the ratio at the point where the K-radiations of tin became prominent. ‘lhis should occur at a potential of about 50,000 volts. No such discontinuit ty appeared. The reason probably lies in the fact that in these experi- ments the tube was fitted with a tungsten anticathode. The applied potential was never sufficient to excite the K-radiations of tungsten to any extent, while the L-radiations would be very largely. absorbed in the walls of the tube and in the window of the examination vessel. Under these conditions, the radiation used was what is generally termed “ white.” Its quality varies with the potential used to excite the tube, but not to avery marked extent. Ulrey* has shown that, under conditions which are very similar to those obtaining in these experiments, there isa maximum X-radiation at a wayve-Jength which obeys the approximate law— i V? = constant, (max.) where V is the applied potential. Thus quite a large difference in the potential produces a relatively small change in the position of the wave-length to which corresponds the maximum X-radiation. It would appear, therefore, that the electron emission associated with the characteristic radiations of the elements bombarded is not of sufficient intensity as materially to affect * C. T. Ulrey, Phys. Rev. xi. p. 401 (1918). 804 Mr. G. Shearer on the these experiments; and these results are to be taken as repre- senting what happens when the characteristic radiations are not excited to any extent. In view of De Broglie’s results, it is to be expected that many of the electrons emitted in connexion with the characteristic radiations will have relatively small velocities, and will, therefore, have difficulty in escaping. from the radiator. Only a ‘small fraction’ of those liberated will escape and contribute to the effect measured in these experi- ments. (7) Selective Emission in the Direction of Electric Vector.— On the classical electromagnetic theory of light it seems probable that there will be a large preponderance of electrons emitted in the direction of the electric vector in the X-ray beam. In fact, the photographs obtuined by C. T. R. Wilson seem to show that suchis the case. In order to find evidence for such an effect, two adjacent sides of the cube were covered with the same metal, and measurements of the electron emission were made when the rays fell perpendicularly and at an angle of 45° onthe metal. In the second case the area of metal under bombardment was 1°4 times the area in the first case, and for this reason an increase of 40 per cent. in the emission is to be expected. When the rays fall perpen- dicularly, electrons emitted in the direction of the electric vector should have difficulty in escaping from the metal, but when the angle of incidence is 45° this difficulty should not be so marked. | The results of such measurements made with lead as radiator showed an increase of 35 per cent. when the angle of incidence was 45°. This increase is rather more than accounted for by the increase of area of the radiator, and these observations show no evidence of any selective emission in the direction of the electric vector. It is probable that by the time that the electrons emerge from the metal their direction of motion is very different from what it was initially, owing to encounters with the atoms of the metal ; and experi- ments such as these could not be expected to throw light on the initial direction of motion of the electrons. Wilson’s photographs were taken with a gas as source of electrons, and in this case the initial direction of the electron is directly observed. (8) Variation of Hlectron Emission with Applied Voltage.— A few experiments were carried out to see how the number of electrons emitted per unit intensity from any one radiator Emission of Electrons by X-Rays. S05 varied with the voltage applied to the tube. Such experi- ments are rendered difficult by the necessity of obtaining some means of measuring the intensity of the X-ray beam. Asa rough measure of this, the ionization produced in the standardizing vessel was used. The electron emission per unit intensity was measured for various potentials ; ‘Table ETT. gives a typical set of such observations. Tei ELE, Electron Emission per Voltage. Unit Intensity. nv?2. 28000 85 14200 36000 81 15400 45000 ral 15000 65000 61 15500 78000 58 14800 This Table shows that the electron emission per unit intensity diminishes as the applied potential increases. Column 3 shows that the product of the number of electrons emitted per unit intensity and the square root of the voltage is approximately constant. Owing to the method adopted for the measurement of intensity, too much reliance must not be attached to this result, but it gives some indication of the nature of the variation. This suggests an intimate connexion between this result and that obtained by Ulrey, to which reference has already been made. Asan approximation the X-:adiation used in these experiments may be considered as a monochromatic radiation of wave-length Ds aay Ulrey’s result, combined with that just given, leads to the result that the number of electrons emitted per unit intensity is proportional to the wave-length, or inversely proportional to the frequency of the exciting radiation. As the size of the quantum is pro- portional to the frequency, this may be interpreted as meaning that the number of electrons emitted per unit intensity is directly proportional to the number of quanta involved. B. Low-Velocity Emission. (9) In studying the properties of these electrons, the total number of electrons escaping from the radiator was measured under the influence of varying, accelerating, and retarding electric fields. or this purpose, the wide-meshed wire cylinder described above was charged to positive and 806 Mr. G. Shearer on the negative potentials, the walls of the vessel being connected to earth. : Under the influence of a retarding field the electron emission diminished rapidly at first, then more slowly, finally reaching a constant value. This stage was reached when a negative potential of about 20 volts had been applied. On the other hand, an accelerating field produced an increase in the emission, and the potential necessary to ensure constancy of the emission was considerably greater than in the case of a retarding field. Fig. 3. ELECTRON EMISSION. — 120 -§0 —40 +120 Cc 0 +40 + 8¢ The curve in fig. 3 is typical of the effects of accelerating and retarding fields on the total electron emission. Such curves were obtained for various metals and for some salts. The resalts of these observations was that there did not appear to be any change in the velocity distribution of these slow electrons with a change in the nature of the matter from which they came. Exact quantitative measurements of the properties of these electrons were rendered difficult by the large effect which the state of the surtace had on the emission. specially in the case of the salts examined, fatigue effects were observed which were presumably of similar origin to those observed in the photo-electric effect. No special precautions were taken to obtain very clean surfaces. ‘The state of the surface has very little effect on the emission of high-speed electrons, whose properties were the main object of this investigation. Experiments made on the ratio of the number of low- speed electrons to the total emission showed that this was also independent of the material bombarded. ‘Table IV. shows some of the results obtained in these measurements. The numbers have been reduced so that they all show the same electron emission with no field acting. Emission of Electrons by X-Rays. 807 TaBueE LY. Voltage. Ou, Sn. Pb. 0 100 100 100 +400 156 140 139 — 400 81 79 80 The results of the following experiment throw some light on the source of these slow electrons. The wire cylinder was removed, and measurements were made on the effect of retarding fields. It was found that, under these conditions, the diminution was much more marked, and in some cases the brass cube even acquired a negative charge with a sufficiently large retarding field. This implied that more electrons were “being driven from the walls into the cube than were coming from the cube owing to the direct action of the X-rays. As precautions had been taken to prevent the X-rays from falling directly on the walls of the vessel, the only sources of electrons from the walls were the high- speed electrons and the scattered and characteristic radiations from the cube. These X-radiations would be of small intensity compared with the direct X-ray beam, and their effect would be still further reduced by the paper lining. On the other hand, the walls were subject to direct bombard- ment by allthe fast electrons, and it seems certain that it was this electronic bombardment which liberated the low-speed electrons. As the energy of the slow electrons is so small, it is reasonable to assume that one high-speed electron can produce a large number of low-speed electrons; and this would account for the observed fact that the number of electrons driven back by the action of the retarding field was greater than the number of electrons liberated by the X-ray beam. With the cylinder in position, the field between the walls and the cylinder prevented the electrons from the walls from reaching the central insulated system, while, owing to the wide mesh of the cylinder, the number liberated from it would be small. The properties of these slow electrons are very similar to hose of the delta rays produced by bombardment by alpha rays. These results indicate that the low-speed emission is of a secondary nature, and owes its origin not to the direct action of the X-rays, but to the high-speed electrons ejected by the X-rays. The process appears to be exactly analogous to the phenomenon of ionization ina gas. ‘lhe photographs obtained by C. T. R. Wilson show that each high-speed electron is 808 Eimussion of Electrons by X-Rays. capable of ejecting a large number of slow electrons from the atoms with which it comes into collision. Exactly the same process should occur during the passage of electrons through a metal. In this case the number which will succeed in escaping will be relatively small owing to their low speed and consequent rapid absorption in the metal. The fact that quite a small accelerating field considerably increases the number of these electrons, shows that many of them are unable to escape unless their energy is increased by outside fields. (10) Summary of Results. (a) The electron emission from various metals and a few salts under the influence of X-rays has been measured, and an attempt has been made to deduce from these measurements the relation between the number of electrons emitted per atom per unit intensity per unit time and the atomic number or atomic weight of the substance from which they are liberated. (6) It has been found that under these experimental conditions the special electron emission associated with the characteristic X-radiations of the substances bombarded is in these experiments only a very small fraction of the total electron emission. (c). These experiments have yielded no evidence of any selective emission in the direction of the electric vector in the X-ray beam. (d) It has been shown that the electron emission per unit intensity diminishes as the penetrating power of the radiation 1s increased. (e) The properties of the low-speed - electrons which accompany the high-speed emission have been investigated, and this emission has been ascribed to a secondary effect due to the action of the fast electrons. This work was carried out at the Cavendish Laboratory, _ Cambridge, and the author has great pleasure in acknow- ledging the continual kindness and many helpful suggestions he has received during its progress from Prof. Sir Hrnest Ruthertord, F.R.S. [ 809 ] LXX. Impact Tonization by Low-Speed Positive H-Ions in fIydrogen. By A.J. Saxton, M.Sc., Assistant Lecturer wn Physics, The University of Sheffield *. Introduction. VHERE now exists a considerable amount of evidence on the conditions necessary to ionize a “normal” atom of a gas. In every case we are concerned with the energy exchanges between the atom to be ionized and the source of the ionizing energy. We may formulate the conditions governing this inter- change of energy as follows :— (a) In order just to ionize a “normal” atomf of a par- ticular gas or vapour, z.e. to detach completely one electron from it so that this electron possesses no kinetic energy as a result of the process (single ionization), always requires the same total absorption of energy whatever the nature of the ionizing source. Thus whether the atom is ionized by electron impact or by the absorption of radiation, the total ionization energy necessary to change it from the “ normal”’ unexcited state to the ionized state is always the same. The long wave-length limit (threshold frequency) for the photo-electric effect in the vapour thus corresponds to the ionization potential for electron impact. (b) “Single” ionization of an atom may be produced by absorption of the ionization energy from one or more of the following sources :— 1. Impact by an electron. 2. Impact by a positive ion. 3. Absorption of radiation (photo-electric ionization). 4. Impact by “ normal” atoms (thermal ionization). (c) If only one of the above sources is concerned, e. g. electron impact (about which we possess the most data), the absorption of the energy may take place in a single process or by stages ; in this case by successive electronic collisions. If a partially ionized atom receives the extra amount of energy necessary to ionize before radiating the portion it has already received, ionization will result. * Communicated by Prof. S. R. Milner, F.R.S. {1 With polyatomic molecules energy may be required first to dis- sociate the molecule. Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3G 810 Mr. A. J. Saxton on Impact Lonization by (d) More than one of the above sources may be concerned in the ionization of an atom. Thus an atom may be partially ionized by absorption of radiation, and then the process may be completed by electron impact *. 2. Ionization by Positive Ion Impacts. The method of studying the ionization produced by one of these four methods alone must be by some form of discharge through the gas at low pressures. The conditions of the experiment must approximate to the following ideal condi- tions. Only the source of ionization, e. g. positive ion impact which is being studied, must contribute to the ionization of the gas, other sources being eliminated as far as possible by suitably designing the apparatus. We must also be able to distinguish between ionization produced by a single process and that produced by the cumulative effect of several colli- sions. Assuming the foregoing principles concerning the energy exchanges in ionization, and given the ideal conditions stated above, we may expect the following result :—That for a posi- tive ion accelerated through an electric field and striking an atom of the gas, in order just to ionize the atom its kinetic energy must be equal to the ionization energy of the atom. ‘Thus if it has fallen unimpeded through an acceler- ating field of V, a singly-charged ion will have a kinetic energy of i Ve=tm’ 2g e This accelerating P.D. of V reduced to volts is the ioniza- tion potential for positive ions. In a p rticular gas it should have the same value as the ionization potential for electron impact. : i This ionization potential for positive ions should be inde- pendent of the nature cf the ions, the effect depending only on their kinetic energy. These principles will apply only to the case where an atom is ionized by the impact of a single positive ion (possessing one positive charge) which has been accelerated unimpeded by a P.D. of V volts. The case of positive ion impact may differ, however, from the case of electron impact in the following manner :— Though the kinetic energy of the accelerated positive ion is the same as that of an electron accelerated through a P.D. of the same value, the energy exchanges with the “ struck ” * H. D. Smyth and K. T. Compton, Phys. Rev. xvi. p. 501 (1920). (Iodine vapour.) | 4 3 | A ee are et Low-Speed Positive H-Ions in Hydrogen. 811 atom may not be so simple, the positive ion itself being a complex system. ‘Thus the ionization potential might depend on the state of excitation ot the positively-charged “ strik- ing”? atom. 3. Lxvperimental evidence on Tonization by Positive ons. To test the validity of these principles, we may examine existing evidence on impact ionization by positive ions, taking inio account to what extent the experimental condi- tions approximate to the ideal conditions for the test. (a) High-speed Positive Lons.—a-rays. Millikan * has shown that ionization by a@-rays usually results in single ionization, more rarely in double ionization. Double ioniza- tion, i.e. detaching the second electron from the already ionized atom, of course requires a greater amount of energy than that necessary for single ionization. Assuming that in the flight of an a-ray through hydrogen there is no loss of energy in non-ionizing collisions, and that each pair of ions produced levies « toll on the kinetic energy of the a-particle of the same amount, and knowing the initial kinetic energy and the total number of ions produced, we may calculate the energy required t» ionize a single molecule. ‘This energy is that acquired by a single charge moving through 35 volts Tf. Since, however, much energy may be wasted in producing partial ionization or in useless kinetic energy of the ejected electron, we may regard this figure as a maximum value for the ionization potential for a singly-charged positive ion. Canal or Positive Rays.—By applying a cross-field of a few volts behind a perforated cathode in a discharge-tube, Stark { used a galvanometer to measure the ionization current produced by the impact of the canal rays on the gas. In nitrogen at a pressure of 0°134 mm. he obtained signs of ionization when the cathode P.D. had a value of 500 volts. These conditions are very different from the ideal conditions necessary. At such high pressures the mean free path of the positive ion would be very small, and therefore we do not know to what extent when they produce ionization their kinetic energy is comparable with that derived from the total P.D. of 500 volts. Moreover, in both these cases of ionization by high-speed positive ions it is probable that the mechanism is quite * Millikan, Phys. Rev. Dec. 1921, p. 446. + Rutherford, ‘ Radioactive Substances and their Radiations,’ p. 159. t J. Stark, Annalen der Physik, 1906, p. 427. See also K. Glimme and J. Koenigsberger, Zevts. fi Phystk, 6, iv. pp. 276-297 (1921). 3G 2 812 Mr. A. J. Saxton on Impact Lonization by different from that operating with low-speed impact where the striking particle does not penetrate the atom. In the case where the kinetic energy of the positive ion is great enough for it to penetrate the atom (a-ray), the amount of ionization per cm. path increases as the speed of the ion decreases. The amount of perturbation increases with the time taken for the ion to cross the atom. Thus a-rays produce most ionization near the end of their range. Glasson’s * experiments on ionization by cathode rays indi- cate a similar effect with high-speed electrons. (6) Lonization by Collision.— Townsend | measured the current between two metal plates with different field-strengths between them, the negative plate being illuminated by ultra- violet light. For small distances between the plates the results could be readily explained on the assumption that the photo-electrons emitted by the negative plate (and the electrons they produced in the gas by ionization) when accelerated through the field produced a-ions per cm. by collision with the molecules of the gas. For distances between the platesabove a certain value he obtained currents which were larger than would be expected on the above simple theory, and ascribed this increase to ionization by positive ions which produced G-ions per cm. For example t, in hydrogen at 8 mm. pressure with a distance between the plates of -3 em. and a field of 700 volts per cm. giving a P.D. of 210 volts between the plates, he obtained an increase ascribed to the action of positive ions. Thus positive ions falling through 210 volts in hydrogen at 8 mm. pressure ionize by impact. The M.F.P. of a positive ion would be very small at this high pressure, so that it could not obtain an unimpeded fall through more than a fraction of a volt. This suggests that the ionization pro- duced may be due to (a) either successive collision or (b) an accelerated positive ion does not lose the whole of its kinetic energy on every collision, and so may acquire a velocity corresponding to a P.D. which is greater than that along its M.F’. path. (c) Cathode Fall in Discharge-tubes.—It is not yet clear whether the positive ions accelerated through the cathode fall of potential produce electrons (cathode rays) by impact with the molecules of the gas or with the metal or occluded gas of the cathode itself§. The values of the minimum * J. L. Glasson, Phil. Mag. (6) xxii. p. 647 (1911). + J. S. Townsend, ‘ Electricity in Gases,’ Chapter IX. t bed. p. 317. § Ratner, Phil. Mag., Dec. 1920, p. 785. Low-Speed Positive H-Lons in Hydrogen. 813 eathode fall of potential are very similar to those obtained for the minimum sparking potential in gases at the same pressure. In hydrogen their values are between 200 and 300 volts. In both these cases ionization by collision of positive ions becomes very important, but the corresponding pressures are so high that in no case will the aecelerated positive ion fall unimpeded through the total P.D. Thus the essential condition for the test is not fulfilled in this case. (d) Positive Thermions from Glowing Filaments.—Stark * measured the current between a glowing carbon filament as anode and a metallic cathode 6 mm. apart in air at °22 mm. pressure with different applied P.D.’s. After obtaining saturation of the positive current for smaller values of the applied P.D., he obtained an increase of the current at 300 volts which he ascribed to the positive ions from the filament producing ionization in the gas. McClelland t obtained a similar result using an incandescent anode in air at -66 mm. pressure, when he found that an increase of current took place at 240 volts. In both these cases the pressure is so high that the P.D. between the two ends of the M.F.P. is only a small fraction of a volt. The only experiments in which the necessary conditions have been at all fulfilled are the following three cases, in which, however, the positive ions were those emitted by glowing coated and uncoated filaments, so that their nature was not known exactly. Pawlow t measured the ionization produced by positive thermions from coated filaments, when accelerated through small potentials, by Lenard’s method. He obtained signs of ionization in hydrogen at as low as 10 volts, and found that the minimum potential at which ionization could be detected varied with the supply of positive ions, being smaller for a greater intensity of the source. He also found that positive ions were much less efficient in producing ionization than electrons accelerated through the same voltages. Thefact that the minimum potential depends upon the original number of positive ions suggests loniza- tion by successive collision. Franck and Eva v. Bahr in similar experiments with air and hydrogen obtained signs of ionization in a gas at potentials below the ionization * Stark, Annalen der Physik, 1906, p. 427. + McCielland, Phil. Mae. xxix. p. 362 (1915). I Pawlow, Proc. Roy. Soc., July 1914, p. 398. § Franck and Eva y. Bahr, Verh. der Deuts. Phys. Gesell. Jan. 1914, p. 57. 814 Mr. A. J. Saxton on Impact Ionization by potentials for electrons in the gas. They concluded that there was no sharp ionization potential for positive ions in the gas, and also found that the minimum potential at which ionization could be detected was lower the greater the intensity of the source of positive ions. Horton and Davies* made a thorough investigation of the ionizing properties of the positive ions emitted from an incandescent tantalum filament in helium. Their results indicated the production of fresh ions by collision of positive ions accelerated through 20 volts. Further investigation led to the view that the ionization produced was not due to the ionization of the gas molecules by direct positive ion impact, but to the bombard- ment of the walls of the ionization chamber by positive ions releasing 6-rays. They conclude that the positive ions do not produce ionization when accelerated through potentials of 200 volts. Thus the only investigations satisfying the necessary conditions are to some extent contradictory. 4. Description of Apparatus. The following experiments were undertaken with the hope of obtaining more definite information about the conditions under which ionization is produced by low-speed positive ions. Other workers have used the positive ions from glow- ing filaments so that their nature was not definitely known. In the present work the nature of the ionizing positive ions was known with greater certainty. They were produced by electron impact in hydrogen, and the speed of the colliding electrous was great enough to dissociate the molecule on ionization. Thus the positive ions formed would be H- nuclei (protons). They were accelerated through hydregen at very low pressure and their ionizing properties studied. In figs. 1 and 2, F is a tungsten filament heated by a battery of 8 volts supplying a current of 2 to 3 amps. and insulated on paraffin-wax blocks. The filament leads are sealed into a glass tube, the end of which fits as a ground- glass stopper into the side tube B. With this arrangement, when the filament burnt out, it could be replaced more easily. The electrons from the glowing filament F are accelerated towards the nickel electrode A by the P.D. of V volts between the negative end of the filament and A. If V is greater than the ionization potential for electrons in hydrogen, ionization by electron impact occurs near to A, - and the resulting positive ions are accelerated towards the - filament by V. Some strike the filament, but some pass on - ™ Horton and Davies, Proc. Roy. Soc., March 1919, p. 383. Low-Speed Positive H-Jons in Hydrogen. 815 through the gauze G,. These positive ions may be further accelerated by the P.D. of y volts between the gauzes, whence they pass into the ionization chamber G,D, where the ionization they produce. can be measured. The nickel gauzes G, and G, are of fine mesh, and are fitted on frames so that they do not touch the glass walls of the main tube. This lessens the possibility of a current leak across the glass. Fig. 1. B pA tee F G, Gs Fig. 2 [hh Ey One set of wires in each gauze is set vertical and the other set horizontal so that the corresponding holes in the gauzes are opposite. EH, and E, are side electrodes for using a cross-field to measure the ionization. Before fitting, the tube was thoroughly cleaned and dried. After fitting to the supply tube, it was exhausted as much as possible by an automatic mercury pump, the glass walls 816 Mr. A. J. Saxton on Impact Tonisaiion by being heated to drive off occluded gases, and the filament was made white-hot. The hydrogen was prepared by electro- lysing a solution of baryta in distilled air-free water. The prepared gas was then allowed to stand for several days over P2205. V,«#>16, which sets aminimum value to the total accelerating field of w+ y. (d) The positive-ion current entering the ionization chamber was measured for different values of « and y and for different pressures (the retarding field being kept at zero) by measuring the rate of charging up of the electrode E connected to the electrometer. For higher pressures (‘1 mm. He) the positive-ion current reaching EH was only a small fraction—about one-tenth—of the original positive-ion current leaving the cylinder. As the pressure was diminished, a greater proportion reached H, indicating that at higher pressures many of the — ions were stopped by collision with the molecules. The whole of the batteries, leads, and measuring instruments were insulated on paraffin-wax blocks, and tests carried out to ensure that none of the measurements of the different currents were affected by spurious leaks. Detection of Lonization.—The first method employed to detect ionization by collision by the positive ions was to take fixed values of x and y and to measure the charging up of H as the retarding field z¢ was increased gradually. No pre- cautions were necessary to distinguish between ionization and radiation produced in Gok. "In the case of electron impact the collecting electrode is negative with respect te the gauze, and so it may charge up positively by releasing electrons under the influence of the radiation, or it may charge up positively as it collects the positive ions. In the present case, since E is positive with respect to Go, radiation will not release electrons from KE, which will collect the negative ions produced on ionization. The ionization currents were measured by the steady potential assumed by the electrometer quadrants when connected to earth across a high-resistance leak. The values of ¢, i.e. the P.D. between G» and H, were corrected for this potential assumed by E. "The type of curve obtained plotting the current to E against an increasing retarding field z is shown in fig. 5. Great difficulty was experienced in keeping the positive ionizing current sufficiently constant so as to render the resale « comparable for different values of ¢. For small values of z the current (potential assumed by HB) was positive, but gradually diminished as z increased until for a certain value of z it became negative, further increase of z resulting in an increase of the negative current. This a ee Cee SL el q Total Current Zo EZ. Low-Speed Positive H-lons in Hydrogen. 821 “cross-over” potential is shown in fig. 5 at 25 volts. The effect of increasing the thermionic current and so the posi- tive ionizing current is to shift the graph along to the right so that the cross-over potential is increased. These facts can be explained readily by the fact that the incoming positive ions ionize by collision. The current to H is thus made up of two currents—the positive ionizing current and the negative current of the negative ions produced. When z is sufficiently large, this negative electron current is sufficient Fig. 5. x = BO lv. y= Avy. Total] Accelleratt eS) O +5 10 15 20 ZS 30 35 40 45 Total retarding Field (Z volts) to swamp the positive current retarded by z and by collision with the gas molecules. In this manner definite evidence of ionization by positive ions accelerated through potentials as low as 18 volts has been obtained. Other ionization tests were carried out with the retarding potential z of constant value and greater than the minimum value of «+y, the total accelerating BaD. fable. [.. wWlus— trates the relation between the positive ionizing current and the negative ionization current for different applied Pye 822 Impact Ionization by Positive H-Lons in Hydrogen. TaBueE I. Total Current ee | ace. P.D. +ive ionizing ae (e+y). | leaving cylinder. Tare C,/C,. (Volts). (Scale Divns.). | [10—-’° amp.]. One Cx 36 +50 9 56 mt +100 15 6:7 92 +60 10 6 + +120 20 6 03 +145 23 6°3 “3 +290 60 4:8 Hydrogen ...... p= 02mm. Hg. V=18 volts. z=100 volts. The positive ionizing current was varied by varying the filament current. The results show that the amount of donization is proportional to the ionizing current. The former current collected under a field of 100 volts is probably increased by the fact that the original electrons released ‘by the positive ions themselves produce some fresh ions by -collision. By obtaining the ratio of the two currents we can estimate a maximum value for the efficiency of the positive ions as lonizers. ‘The ratio of the ionization current to the ionizing current for the results in Table I. is about 1/130. Thus of at least 130 positive ions reaching ihe ionization chamber G»2H, only one ionizes a molecule cf hydrogen. Electrons accelerated through the same voltage -and at the same pressure in hydrogen would produce about ‘one positive ion per ionizing electron*. Hence when moving with these low velocities, the efficiency of an electron is much greater than that of the positive ion in producing ionization by collision. Most of the energy of the positive ‘ions is lost probably in non-ionizing collisions. Summary. An attempt is made to formulate the conditions governing ‘the exchange of energy when an atom is ionized by impact -or by the absorption of radiation. This is applied to the case of ionization by positive-ion impact with the conclusion that the ionization potential for positive-ion impact in a gas * J.B. Johnson, Phys. Rev. 1917, p. 609. On the Partition of Energy. : 823 should be the same as that for electron impact. Results of previous workers show wide disagreement in the value of the minimum accelerating P.D. necessary for positive ions to produce ionization. Two types of apparatus are described, the positive H-ions being produced first by electron impact i in hydrogen. The peculiar difficulties of the experiment made it impossible to test for ionization produced by positive ions accelerated through less than 18 volts, and, as the ionizing current itself depended upon three variable factors, it was more difficult to obtain comparative observations. The results indicate that positive H-ions produce ionization in hydrogen when accelerated through P.D’s. as low as 18 volts, and that they are much less efficient in producing ionization than electrons under similar conditions. In conclusion, I desire to thank Prof. Milner, F.R.S., who proposed the problem, for his kindly interest and advice. Sheffield University, July 31st, 1922 ———— LXXI. On the Partition of Energy.—Part II. Statistical Principles and Thermodynamics. By C. G. Darwin, F.RS., Fellow and Lecturer of Christ’s College, Cam- - bridge, and R. H. Fow er, Fellow and Lecturer of Trinity College, Cambridge * § 1. Introduction and Summary. . a previous paper T we have-developed a method of calculating the partition of energy in assemblies containing simple types of quantized systems and free molecules. In this method advantage is taken of the multinomial theorem, which makes it possible to obtain integrals expressing accurately the various average values concerned in partition problems, and then the ‘integrals are easily evaluated asymptotically. The “ fluctuations ” can also be readily found, and from them it can be shown that the possession of these mean values is a “normal” property of the assembly in the sense used by Jeans f. This method has the advantage of obtaining directly the * Communicated by the Authors. + Phil. Mag. vol. xliv. p. 450 (1922). ¢ Jeans, ‘ Dynamical Theory of Gases,’ passim. Average and most probable values are of course in practice the same. 824 Messrs. C. G. Darwin and R. H. Fowler on average properties of the assembly, which are, on any statistical theory, those which experiment determines, instead of the most probable values, as is usually done. It can also be carried out simply and rigorously without the use of Stirling’s theorem, and thus provides satisfactory proofs of all the usual partition laws, including Maxwell’s Distribution Law. In this discussion the partition laws were all obtainable without any reference to thermodynamical ideas, in par- ticular without any mention of entropy. This we claim as an advantage. But a great deal of work has been done on partition laws, in which the idea of entropy has played a leading part; so that, for this if for no other reason, it is fitting to examine its position in our pre- sentation of statistical theory. But the power of our method on the statistical side invites a somewhat more general review of the fundamental connexion between ciassical thermodynamics and _ statistical mechanics both of classical dynamics and the quantum theory. In the former work we were content with purely statistical results, and identified the temperature scale simply by the perfect ° gas laws; here we attempt a more strictly logical deve- lopment, and prove the Jaws of thermodynamics for assemblies composed of systems of a fairly general type, and, by linking on to Gibbs’ work, also for general systems - which obey the laws of classical mechanics. After summarizing our previous resuits in § 2, we pass in § 3 to a comparison between the empirical temperature in thermodynamics and the parameter which acts as temperature in our previous work. In §§4, 5, 6, we make a critical study of the usual presentation of entropy in statistical theory. This is ordinarily introduced by means of Boltzmann’s Hypothesis, which relates it to probability, and, though no objection can be made to much of the work based on this hypothesis, it appears to us that the development is often marred by somewhat loose reasoning. Though much that we here say is general and not at all dependent on our special methods of treatment, yet it has been far easier to examine the validity of the arguments on account of the way in which it is possible to combine assemblies together at will. In con- sequence of this discussion we are led, in §7, to a pre- sentation of entropy which is very closely related to that of classical thermodynamics, which frees it from the com- binatory complications with which it is normally associated and brings it back to direct dependence on the partition the Partition of Energy. 82 5 functions which form the basis of our method. In § 8 the definition is considerably simplified mathematically by replacing the “entropy ” by the “characteristic function as the basal thermodynamic quantity. In §9 we show that for an assembly in a temperature bath our method is exactly equivalent to Gibbs’ use of his ‘canonical ensemble,’ and in § 10 we consider briefly the possibility of inverting the argument so as to obtain information about the elementary states from thermodynamic data. 99 § 2. Summary of our previous paper. It was shown in the previous paper how the partition of energy could be evaluated for various types of assembly. Those chiefly treated were quantized systems, for which the energy was sole variable ; but it was also shown how to apply the method in the case of a perfect gas, where both energy and volume are variable. It is easy to see that the method is applicable in considerably more general cases. The partition of energy could be evaluated when any of the types enumerated were mixed together, the essential point of the method being the existence of a certain function, the partition function, for each type of system. By means of these functions all the rather tiresome combinatory ex- pressions can be very easily dealt with so as to obtain mean values, and also the fluctuations about those mean values. The partition function of asystem—which with a different notation is the “ Zustandsumme ” of Planck—is constructed as follows. The possible states of the system may be divided into cells; these cells are fixed and finite for quantized systems, but for the systems of classical mechanics must ultimately tend to zero in all their dimensions. Associated with each cell is a weight factor, determined by the usual statistical principles. The weight of any cell of a system obeying Hamiltonian equations is proportional to its ex- tension, The relative weights of the cells of a quantized system are determined by Bohr’s Correspondence principle, and the weights are all assigned definite magnitudes by the convention that a simple quantized cell shall have unit weight. For consistency in physical dimensions the cells for Hamiltonian systems are divided by the appropriate power of fh to give their weight. Associated with any cell there is a definite energy, depending on the cell and on certain external parameters 7, w2,...3 this last isa slight extension of our previous paper, which must be made so as to deal with questions of external work. Then, if p, is the Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 23H 826 Messrs. C. G. Darwin and R. H. Fowler on weight and e, the energy in the rth cell, we define as the partition function i = Spe oa summed over all the cells. The partition function is thus a function of 3, x, 2, ..... For systems of classical mechanics the extensions of the cells, that is the p’s, all tend to zero, and the sum is replaced by an integral. As an example of the systems treated we may mention first Planck’s line vibrator, which can take energy in multiples of-e. This has the partition function ; fis: (2:1) More important is the free monatomic molecule. If it is of mass m and is confined in a volume V, its partition function is (2am)??? V hog 1/3)?” (2-2) The partition function of a diatomic molecule is simply given by multiplying (2:2) by the partition function corre- sponding to the quantized rotations (assuming that the atoms cannot vibrate relatively to one another). Now suppose that we have any number of types of systems together in an assembly ; let there be, say, Ma of type A with partition function fy. Then it was shown — that the average energy among the systems of this type was given by = fo Ky = Mas sy log fa, nuts Aa: (2°3) where 3 is uniquely determined in terms of E the total energy by 0 It is easy to show that 3 is always less than unity. The average number of systems A in the rth cell is also easily found, and is Gea My nyo Sanifo. 0 Sl In connexion with the relation of entropy to probability, we must also recapitulate some of the work from which the above results were derived. The statistical state of an the Partition of Energy. §27. assembly, say of two groups A and B, M and N in number respectively, is specified by giving sets of numbers dp, a, Megs io Oy. bj seo, where .a./is the number of :the “A's which lie in the sth cell. Suppose that this cell has a weight factor p, and that the sth cell for a B has a weight factor gs: then the fundamental basis on which the whole of statistical theory rests gives an expression, “M1 om N! by By as 2 : 96) a —————— hi) 1 date enoe Go G1 LL Cte 2°6 reentia.f) awe meee Ie esta ka0) for the number of ‘* weighted complexions ” corresponding to that specification. If this number is divided by the total of all the weighted complexions, which we call OC, which correspond to any distribution of a’s and 0’s con- sistent with the same total energy, then this ratio is the proper measure of the probability of the specification, and must be used in calculating the expectation of any quantity. It was shown in the former paper that C and the associated averages can be expressed as contour integrals (exact for quantized systems) which lead asymptotically to the formulee (2°3), (2°5). § 3. Temperature. In considering the connexion between statistical theory and the principles of thermodynamics, we must begin by correlating the ideas of temperature in the two theories. Throughout our former paper we have treated the para- meter 3 as of the nature of the temperature, and it is here of some importance to observe that 3 has precisely those properties which must be postulated of the ‘‘ empirical temperature ” when the foundations of thermodynamics are rationally formulated *. The basal fact of thermodynamics is that the state of two bodies in thermal contact is deter- mined by a common parameter which is defined to be the empirical temperature. The temperature scale is at this stage entirely arbitrary, and any convenient body whatever may be chosen for thermometer. Cn the statistical side we have shown that when two assemblies can exchange energy so that there is one total energy for the two together, then their states are defined by a common para- meter 3. The analogy is exact, and we are therefore logically justified in identifying 3 with the empirical temperature in precisely the thermodynamical sense. * See, for instance, Max Born, Phys. Zeit, vol. xxii. pp. 218, 249, 282 (1921), ale 2 828 Messrs. C. G. Darwin and R. H. Fowler on Since we may take any body for our thermometer by which to measure the temperature scale, we may if we like at once define the absolute temperature of statistical theory as proportional to the pressure in a body of perfect gas at constant volume. Now if there are P molecules in volume V, it follows from almost any theory that lah ai Be x7 3 Chins; Merete hee (0 |. where @,in, 13 the mean kinetic energy of translation of a molecule. In terms of 3 we have by (2°2) and (2:3) ie e log 1/3” Y 3 Pein: — 2 and so we must have log L/S’ SL /kT, S's os eee where & 1s a universal constant. This appeal to the properties of an ideal substance is, however, not quite satisfactory. It is avoided in theimo- dynamics, where the absolute temperature is defined in connexion with the Second Law. Now we wish to show how our theorems lead to the laws of thermodynamics, and so we must not postulate a knowledge of absolute tempe- rature, but must only consider it in connexion with entropy. Our development of the Second Law will not of course have the complete generality for classical systems of such treatment as Gibbs’ (though we have later extended it to his case), but still it will suffice to deal with assemblies as general as those that have been used by most writers who have deduced special thermodynamical conclusions from statistical premises. | § 4. The usual presentation of Entropy. Entropy is usually introduced into statistical theory by means of Boltzmann’s Hypothesis relating it to probability. This hypothesis is based in general on the fact that, on the one hand, an assembly tends to get into ils most probable state, while, on the other, its entropy tends to increase, and so a functional relation between the two may be postulated. The general line of argument is somewhat as follows *:— We can assign the numerical value W, for the probability of the state of any assembly. If we have two such assemblies * Planck’s classical work on Radiation Theory is a representative example of the use of the argument here quoted. | the Partition of Energy. 829 which are quite independent, then by a fundamental principle of probability, the joint probability is the product of the separate probabilities ; that is, WW; = Wei. ° . ° ° . ° (4:1) Ou the other hand, the joint entropy is the sum of the separate entropies, and so Sit S. => eee . ° mo ae e ° (4°2) Then to satisfy the functional relationship we must have = OE Wy Sour sw ntny (AS) k being a universal constant. Next, to evaluate W, a definition is made of ‘thermodynamic probability ” as the number of complexions corresponding to the specified state : this is made a maximum subject to the condition of constant energy, and the maximum of & log W is equated to the entropy , denotes summation over all different values ot the 8’s such that Lensls =k — > € ys and of course, as always, ,a,=M, >,bs=N. Now, provided that N is much larger than M, the factor | at | eatin Ks will be practically independent of the a,’s and of the energy of the group of A’s—that is to say, it may be taken as constant and omitted from the calculation, and we are left with the ‘‘thermodynamic probability ” as the only variable art. ; It is only in this sense that a strict meaning can be assigned to Boltzmann’s “Hypothesis; and it is of the greatest interest that the conditions under which it has meaning correspond exactly to the conditions of the ‘canonical ensemble” of Gibbs, as will be shown later. But, even so, it is not a very convenient expression, for we must always suppose that the assembly is a part of some much larger one, whereas the expression for the entropy is purely a function of the group and the tempe- rature. It is therefore more convenient to abandon the use of the principles of probability and to define the entropy us k times the logarithm of the number of complexions (weighted if necessary). We shail call this the kinetic entropy. This number of complexions has the multiplicative property (4°1), but now in virtue of its own combinatory formula and not of an appeal to an inapplicable probability theorem. The new definition does not appear to have the same simplicity as the old, but that is only because in the old the necessity for a detailed definition of what is meant by probability was concealed. It would appear that CO 832 Messrs. C. G. Darwin and R. H. Fowler on some such argument as this is necessary to justify the use of “thermodynamic probability,” the quantity used with success by so many writers. The argument of this section has really been dealing entirely with the junction of assemblies which had the same temperature ; it may be more conveniently visualised as dealing with the separation of an assembly into parts which are thereafter isolated from one another. Actually of course our work must include the fact that entropy has the property of increasing when assemblies at different temperatures are joined. We have not yet had cause to discuss this, as we have so far been mainly criticizing theories which were developed by considering only assemblies of the same temperature. | : $5. Entropy as a non-fluctuating quantity. The kinetic entropy as defined above is a fluctuating quantity, whether we find it for the whole assembly or for a part. On the other hand, the entropy of thermo- dynamics is a function of the state of the assembly and must be regarded as constant, and we must see how the two may be best related. Now we cannot get away entirely from the question of fluctuations, but we can conveniently simplify the definition so as to dissociate them from the entropy. Consider an assembly composed of A’s and B’s. At every moment its state is specitied by the values of CANA ERC ae OFS arn , and these numbers all fluctuate, and with them the energy Ey, and the kinetic entropy Sa. But if we want to treat of the entropy of the A’s as opposed to that of the b’s, we must suppose the A’s to be suddenly isolated. After the isolation they will have a certain definite energy determined by the chance state at the moment of isolation, and this energy will determine the temperature and so the thermodynamic entropy. So, to define a function representing the thermodynamic entropy, it 1s most reasonable to choose some simple non-fluctuating function of the state of the whole assembly; we can then allow for the fluctuations in the entropy of its parts by imagining them suddenly isolated, and calculating their entropies from the energies they chance to have at the moment of isolation on the same principle as was previously done for the whole assembly. There are several suitable definitions—for example, we can use the total number of the complexions, or the average number, or the maximum : | 7 ee 2 a ‘ the Partition of Energy. 833 number, in each case attributing to the A’s the amount of energy they had at the moment of isolation. Now if these quantities are calculated, it will be found that, to the approximation (4°5), they all have the same value. This value is easiest to find for the maximum number. It is unnecessary to take an assembly of systems of more than one type, as we have seen ‘that the additive property will hold. We have S/k = log M!—%, log a,!+ 3,4, log py. We must here make the unjustified application of Stirling’s theorem to numbers some of which will un- doubtedly be small; it should be possible to justify the process, but we shall not do so. Then, making use of (2:1), (2°5), we have | | eae S/k = M| log f—log $.9-S log /'], es ee 2M log fe Milos (1 /ayye es sale ee ABB) since i fo = Ms 53 log f. Equations (5:1) and (5:2) remain equally true for a group of free molecules :to the same approximation. ‘This formula for 8 is the direct consequence of Boltzmann’s Hypothesis, and 8 has the necessary additive property for combining the parts of the assembly. Moreover, it agrees completely with the entropy of thermodynamics in all cases where they can be compared: this agreement justifies our use of (4:5) in these calculations. But it is indifferent whether we define the entropy as the total, average, or maximum number of complexions, and (4°5) is always inexact ; it is therefore unsatisfactory to make the formal definition of non-fluctuating entropy in any of these ways. ‘Now (5:1) and (5°2) give precisely the thermodynamic ex- pressions in all comparable cases, and this suggests a direct definition in terms of partition functions. We may thus suppose that the combinatory processes are correctly looked after by the partition functions, and may define the entropy by either of the relations (5:1) or (5°2). Pending its formal identification with the entropy of thermodynamics, we shall describe it as the “ statistical entropy.” 834 Messrs. C. G. Darwin and R. H. Fowler on § 6. “ The increasing property” of Entropy. We have now obtained a quantity S,,, the statistical entropy, which is evidently related to the entropy of thermo- dynamics Sy, but we must examine what right we have to. make the identification complete. By its definition (5:1),. Sst. has the additive property for separation, and we can easily show that for Junction it has the property 8, + 8. < So). which may be called the increasing property. Consider the special case of two assemblies, and suppose that in their junction only changes of temperature are con~ cerned—not of volume or any other parameter. We shall also simplify by supposing that in each assembly there is. only one type of system, different for the two. As we do. not intend to base our final result on the present paragraph, this wiJl be general enough. By definition, for the first. assembly before junction the entropy is given by Sst. /k = M' log f'(8')—E’ log 9’. Now when the energy H’ is given, the temperature 3’ is determined by (2°4), and this is equivalent exactly to the condition that Ss’ should be a minimum for given HE’. So,. if S$ has any value different from $', Sst. /k —-008473w°—-006938w7], . (19) Vo= — “< [1°750—-053750e —-028650w?—-019046% —-0138884w! —-010613w> ~*008439w°—-006839w"], . (20) while the total energy W is T,+ Vo. | Check on Accuracy of Solution. One of the standard methods of checking the accuracy of computations in Astronomy is to compute the energy and see if it remains constant. This method could be used in our problem, but would involve the calculation of the coefficients of the various periodic terms in the Fourier expansions of T’ and V, which would be extremely laborious, as over twenty pages of computations are required to determine the constant term alone. A much easier method of checking is furnished by the fact that in motion under the 864 Mr. J. A. Van Vleck on the normal Helium Aton inverse square law the average absolute value of the potential energy is twice the average kinetic energy *. Since the average value is simply the constant part of the Fourier expansion, and since a power series development is unique,- the coefficients of like powers of w in the bracketed power _ series in equations (19) and (20) must, therefore, be identical if the computations are correct. There is absolute agreement in the first three terms, while the small errors in the fifth decimal place in subsequent terms are insignificant, and due mostly to neglect of third and higher powers of the perturbations. Parr III.—APppPuLicaTIoN OF QUANTUM CONDITIONS To Mopet with AXIAL SYMMETRY. The same value for the energy is given consistently by several different types of quantum conditions, viz., the value _ obtained by choosing the constants of integration (p and w) so as to satisfy the relations Noh = = g e e « e es e (21) = h cs where n, and ny are integers, ‘I’ is the average value of the kinetic energy (equal to the negative of the total energy), and v1, v, are the two intrinsic frequencies of vibration, given by T y= es the frequency of vibration of the coordinates 7 and z, on q y Vo = me , the frequency of rotation of the cyclic coordinate 0) ne (i. e., ae lites the mean angular velocity of the electrons us about the axis of symmetry). ..9:° 2 9) 9..¢5)) -) ae Equation (21) states that the resultant angular momentum * For proof of this relation, see Sommerfeld, ‘Atombau und Spektrallinien,’ 2nd ed., p. 472. Sioa + For proof that the », and v, defined in (23) are the intrinsic frequencies in the Fourier expansion of the Cartesian coordinates ~, y, 2, see Bohr, ‘ The Quantum Theory of Line Spectra,’ p. 38. and its Relation to the Quantum Theory. 865 7 p must be an integral multiple of In the actual numeri- hr oar” cal determination of the orbits so as to satisfy (21) and (22), T and v, were evaluated as the constant terms in Fourier expansions. The quantum numbers 7 and ny were each taken equal to unity to give the normal orbits, those of lowest energy. Tie value of w was found by trial and error to be *7216, giving an .energy of 74°9 volts, already discussed. The remaining pages will be concerned with showing that various theories devised for quantizing the stationary states demand that equations (21) and (22) be satisfied. Sommerfeld Quantum Conditions. From the standpoint of the Sommerfeld conditions (viz., that \ pidgi= nih) the result (21) is obtained by assuming that the cyclic coordinate @ together with its conjugate momentum p satisfy a quantum integral, so that 2Q0r | pdd=nzh. 0 This is in agreement with Hpstein’s theory that when partial separation of variables can be effected in the Hamilton-Jacobi equation, the Sommerfeld conditions should be satisfied by the phase integrals associated with the coordinates which can be separated™ (i. e., \ pdqi=nih for the particular values of i for which p; may be regarded as a function of g; only) ft. . ; i ’ Also, as mentioned by Bohr {, the value 5 for the resultant Us angular momentum appears to be demanded by the con- servation of angular momentum, independently of quantum theory considerations. For a conditionally periodic system with any number of * Verh. d. D. Phys. Ges. vol. xix. p. 127. 7 It is interesting to note that Epstein’s conditions demand that the resultant angular momentum of any three body system, and hence ofany model of helium (not necessarily in the normal state) be equal to an ‘ ; h ; integral muitiple of =~, for in this much more general case the resultant aT angular momentum can be proved conjugate to a cyclic coordinate of period 27. (For proof, see Whittaker, ‘ Analytical Dynamics,’ p. 345,) t ‘The Quantuin Theory of Line Spectra,’ p. 35 (Mémoires Dan, Acad. 1918), 866 Mr. J. H. Van Vleck on the normal Helium Atom degrees of freedom *, which has two intrinsic frequencies of - vibration vy; and v2, and in which separation of variables can be effected, the Sommerfeld quantum conditions demand that the average value of the kinetic energy be that given in equation (22) f. The general type of motion in the par- ticular dynamical system we are considering is presumably not conditionally periodic, but, instead, the great majority of orbits seem to be characterized by large perturbations, in which the radius may tend steadily, though very slowly, to the value zero or infinity. For this type of motion the Sommerfeld quantum integrals have no meaning (except in case of the cyclic coordinate ¢) and no technique appears to have been devised for quantizing the general orbits in dynamical systems of this character. However, the simple relation given in (22), though not often mentioned in the literature, is one which is satisfied in practically all cases in which quantum theory dynamics have been applied successfully, and consequently may itself be regarded as a quantum con- dition of considerable generality. Therefore, when particular classes of orbits can be found which are conditionally periodic and characterized by two intrinsic frequencies of vibration, one would expect this relation to be satisfied. This. amounts to saying that, since orbits characterized by con- — stantly increasing perturbations cannot occur in the normal state, we need quantize only the families of orbits which are conditionally periodic, which contain two intrinsic frequencies and four arbitrary constants (two of which are epoch angles), and which therefore resemble the general motion in a con- ditionally periodic system with two degrees of freedom f. * If the number of degrees of freedom exceeds two, the motion is partially “ degenerate.” t To prove this the case we observe that by Kuler’s therem on homo- geneous quadratic functions } pt NOR ts Gi le, uO D} Ogi Vi=sPi N, Since Roe Ogi Oa gees eh fb The relation T= 5 (n,7,+n,v,) is obtained immediately by taking time average and using the facts that \ p,4q,=n,h and that the frequency of q, is either », or p,, { If the Poisson and secular terms in which the time appears explicitly should prove to combine in such a way as to make the general motion conditionally periodic, then, if separation of variables could be effected, the Sommerfeld quantum conditions could be applied directly and the general motion could be specified with the aid of three angle variables. The relation (22) would then be obtained by equating to zero the quantum number 7, associated with the third intrinsic frequency », not appearing in the particular family of orbits studied in solving the dynamical problem. and its Relation tu the Quanium Theory. S&T Schwarzschild Angle Variables. The “angle variables ” (Winkelkoordinaten) are intrinsic coordinates which are 27t times the frequencies of vibration of the system, and thus possess the characteristic properties of being linear functions of the time such that alteration of any one of them by an amount 27 leaves the configuration of the dynamical system unaltered. The two angle variables for the family of orbits given in equations (17) and (18) are therefore 27p,t and 2av.t, where vy; and vy have the values given in (23), The intrinsic momenta P; and P, conjugate to the angle variables Q, and Q, are constants defined by the canonical equations dQ a oH é AQe Gs oH ee OP, ae Pi where H is the Hamiltonian function (7. e., the energy regarded as a function of P; and P,). The general solution of the above equations can be shown to be * 1 ee = aa 2m[ R? == Z? \dt, 0 7T 1 = aI pdb=p. 0 Equations (21) and (22) follow immediately on setting P, and P, equal to integral multiples of -- in accordance with Schwarzschild’s quantum conditions, which demand that 27 frag, = nih ° 0 Ehrenfest’s Adiabatic Hypothesis. Ehrenfest’s adiabatic hypothesis states that motions “allowed ” by the quantum theory are transformed into new “allowed” motions as the character of the dynamical system is altered by changing very slowly some parameter appearing in the energy. We shall take this parameter a proportional to the perturbative force of repulsion between * For proof, use methods of Epstein (Ann. d. Phys. vol. li. p. 168). 868. Mr. J. H. Van Vleck on the normal Helium Atom the two negative electrons, so that the total potential energy V of the system is | ar 4¢* OG! : =— JRE + on (cf. equation (2)). For a=0 there is no mutual action between negative electrons, and then each electron describes a circle or ellipse character- istic of a central force obeying the inverse square law. Since the motion in this undisturbed system is periodic, the ae integral must be an integral multiple of h, so that we ave T 2 ia (ny +n9)h, : . 5 co . (24) 0 . where 7 is the period. Also we shall assume that the resultant angular momentum about the axis of symmetry is an integral multiple oe which apparently is demanded by the conservation of angular momentum, and which is required if Sommerfeld’s quantizing of space in polar coordinates is accepted. Now let the parameter a be increased from the fictitious value zero to the actual value unity. Since there are no forces operative which have a moment about the axis of symmetry, the resultant angular momentum retains its =. and the axial symmetry is preserved. Also if we assume that the motion always remains condition- ally periodic as the perturbing field is thus gradually increased, then, using an equation given by Ehrentest *, it is readily shown that the average kinetic energy has the value demanded by (22). original value Bohr’s Quantum Conditions. By quantizing the perturbations in a manner analogous to that of the Sommerfeld conditions for conditionally periodic motions, Bohr has devised a general method for determining tne “allowed ” motions whenever the perturbing potential has axial symmetry f, although his treatment is intended primarily for cases where the departures from the undisturbed orbits are so small that only first powers in the perturbations need be considered. These quantum conditions demand that * Ann. d. Phys. vol. li. p. 348, equation (m). + ‘The Quantum Theory of Line Spectra,’ pp. 53-6. and its Relation to the Quantum Theory. 869 the angular momentum about the axis of symmetry be an ee h eee integral multiple of ;—*, and that the motion be adiabati- at eally derivable from an unperturbed orbit for which the action integral has the value given in (24). Bohr’s conditions are therefore in agreement with the application of the Khrenfest adiabatic hypothesis given above. Jefferson Physical Laboratory, Harvard University. March 13, 1922. Note-—Since this paper was written, an article by Bohr has appeared in the Zertschrift fiir Physik, vol. ix. (1922), p. 1, in which he conjectures that the Kemble model with axial symmetry and with crossed orbits (studied in the present paper) may be the correct solution of the normal helinm atom. It is therefore to be regretted that calcula- tion has given an ionization potential of 20°7 volts instead of 25:4 demanded by experiment. According to Bohr the normal helium atom is capable of formation from a free electron and ionized helium atom by continuous transition through a series of intermediate orbits. The family of orbits in the present by varying the constants of integration p and w furnish a means of transition from the Kemble model to a stationary state of lower energy given by h : : w=0 and ea which gives the coplanar circular orbits of the original Bohr helium model. According to Bohr (p. 32) this mode] cannot be formed by a continuous transition from the stationary states found in the orthohelium model (coplanar orbits of unequal size), but the statement which I have just made makes it appear capable of formation by continuous transition from the stationary states of the parhelium series. The instability which may result from the possibility of degeneration into coplanar orbits of lower energy makes it plausible that the normal state of the helium atom may not be characterized by crossed orbits with axial symmetry. | * Bohr’s conditions demand that the resultant angular momentum of . Noh mts a single electron about the axis of symmetry be = , while in our con- ditions this value was taken for the resultant angular momentum of both electrons, a quantity twice as great. This, however, is probably not a contradiction, as Bohr’s method was derived for systems with only one- electron, SE Seto | wi4 Cc | be sn0 ] LXXIII. The Use of a Triode Valve in registering Electrical Contacts. By G. A. Tomiinson, B.Sc.* fs eee electrode valve can be applied with advantage to certain forms of apparatus in which use is made of electrical contacts. A common case is that of a relay in which it is usual to cause a feeble movement of one instrument to make a contact and close the circuit of a second compa- ratively powerful instrument supplied from an independent source. An improvement in several respects can be made if the first contact is placed in the grid circuit of a valve, and the second instrument is connected in the anode circuit and is operated by the anode current. An arrangement used by the writer is shown in fig. 1. —-------y5 eee ANODE CURRENT. FIG | A small tongue of platinum C is moved by the first in- strument between two platinum points p; and ps, and on making contact either raises or lowers the potential of the grid by about 4 volts by introducing the batteries b, or by. The reaction on the anode current, which is illustrated by the diagram of the valve characteristic, operates the in- strument D. Thus the points P,; and P, show the anode current when the tongue is in contact at p; and po, the dif- ference being the range of current available, which is about — 3 milliamperes. ‘The actual values of the grid potential are adjusted by the battery 6 to vary between —2 and +6 volts, to obtain the best range of anode current. In this particular case the relay is required only to detect small angular movements of an instrument in either direction from the zero position, and the instrument used in the anode circuit is a pivoted moving-coil galvanometer. * Communicated by the Director of the National Physical Laboratory. Use of Triode Valve in registering Electrical Contacts. 871 The resistances 7, and vr, in the circuit enable a fine adjustment of the points p; and p, to be made. ‘These points can be advanced by micrometer screws until both just touch the platinum tongue. This contact can be ob- served by temporarily inserting the galvanometer G in the circuit. The E.M.F.’s of the batteries b, and 6, act in the saine direction in this local circuit, and with suitable re- sistances 7 and 7, a small local current flows when p; and pz are both in contact. The points can then be separated by any desired amount. The resistances 7; and 7, also prevent a short cireuit if an accidental contact is made from p, to po. A relay of this type has several advantages. The current to be transmitted through the contacts is very small, being only the grid current of the valve. Variation in the re- sistance of the contact within wide limits has no effect on the action, owing to the great resistance already in the circuit between the grid and the filament. The amplification of mechanical power, which is the function of a relay, is pro- vided for by the electrical amplifying properties of the valve. A further advantage, that may be important in some cases, is that this relay may be operated by much weaker forces than could be used with an ordinary relay. In the latter type the contact is placed directly in the circuit of the second instrument, and a certain contact pressure 1s necessary to ensure the passave of sufficient operating current. With the valve relay an extremely light contact between clean platinum surfaces is sufficient to charge the grid of the valve, and very weak forces will therefore work the relay in a satisfactory way. Thus it has been found that a contact force of 0:000001 grm. is quite sufficient to charge the grid and produce the required change in the anode current. Since there is practically no current transmitted by the contacts, there is no objectionable coherence of the surfaces, and the movement of the contact tongue can be reduced to a very small amount if desired. For example, the relay has been operated with the travel of the tongue only about 1/1000 mm. Certain modifications to meet different requirements may be suggested. If the contact is for any reason intermittent, a comparatively steady current may be obtained in the anode circuit by connecting a suitable condenser across the grid and filament. A high- resistance grid-leak may be used if it is desirable for the anode current to assume its normal value immediately the contact in the grid circuit is broken. The writer has also made some experiments, using an electrical contact in the way described for quite a different 872. Mr. EH. A. Milne on Radiative Hquilibrium : purpose, namely as an indicator for precise measurement. A compound lever with a magnification of about 600 was arranged to be moved by a micrometer at the one end, and earried the contact at the other end. Using this to measure the thickness of a parallel slip-gauge with the lower face resting on three steel balls and the contact on the upper face, it was found that repetition of observations could easily be obtained with variations not exceeding 0°5 x 106 inch. These experiments indicate that the advantages obtained by making contact in the grid circuit of a valve may eliminate some of the difficulties hitherto experienced in this method of measurement. This method appears to have advantages in connexion with the reception of feeble wireless signals with the aid ofa relay ; and it is also proposed to try it, on account of its freedom from sparking at the grid-circuit contact, in connexion with the location of the height of the mercury surface in the vacuum space of a standard barometer. July 1922. LXXIV. Radiative Equilibrium: the Insolation of an Atmo- sphere. By Wi. A. Minne, M.A., Fellow of Trinity College, Cambridge *. § 1. Iwrropvcrron.—The generally accepted theory of the existence of the earth’s stratosphere was put forward in 1908 by Gold+. Gold showed that when radiation processes were taken into uccount the continued existence of an adiabatic gradient to indefinitely great heights was impossible ; for the upper pertions of such an atmosphere, being very cold, would radiate very little, but on the other hand, being backed by an extensive cushion of warmer air besides the warm surface of the earth, would be subjected to low-tem, erature radiation of considerable intensity, and the consequent excess of absorp- tion over emission would raise their temperature and so disturb the adiabatic gradient. Such upper portions, however, could not exchange heat with the rest of the atmosphere by con- vection, for they would tend to rise, not fall. Consequently * Communicated by the Author. + “The Isothermal Layer of the Atmosphere and Atmospheric Radiation,” Proc. Roy. Soc. 82A. p. 43 (1909). A preliminary announcement was made at the British Association meeting in 1908; see ‘Nature,’ vol. Ixxvili. p. 551 (1908). See also Geophysical Memoirs, No. 5 (Met. Office), vol. i. p. 65 (1913). the Insolation of an Atmosphere. 873 their temperature would continue to increase until the extra emission due to increased temperature balanced the absorption and a new steady state set in—a state of radiative equilibrium. The direct absorption of solar radiation is small and, though important, does not affect the argument. (It is of interest to mention that exactly the same course of argument shows that even in the absence of convection a strictly isothermal atmo- sphere is impossible ; for the outer portions would not be able to absorb as much as they emitted, and so would cool, causing convection. ) Gold embodied these ideas in analysis, in order to determine the temperature and the height of the tropopause, and he showed that the theory generally was adequate to account for the observed values. His procedure, however, was in part empirical. In the light of Schwarzschild’s * theory of radiative equilibrium in a stellar atmosphere, an immediate rough evaluation of the boundary temperature is possible ; if I) is this temperature, then T)*=4T,*, where T, is the effective temperature of the system (earth plus atmosphere) as determined by the amount of energy radiated away into space. This energy is equal to the mean value of the absorption of solar radiation, assuming that the earth is on the average neither losing nor accumulating energy. The value of Tj, deduced by Abbot f from the solar constant and the earth’s albedo, is about 254°, giving Tp =214°. The observed mean value of the temperature of the str atosphere over the British Isles is about 219°. Schwarzschild’s formula, To =1T,*, was indeed obtained independently by Humphreys t in this con- nexion, and applied to the stratosphere. Gold, however, did not proceed in this way. Accepting the observed division of the atmosphere into two shells—an inner one in convective equilibrium with a known temperature gradient, and an outer one at a uniform temperature,—he determined the height at which the convective gradient should terminate, in order that the atmosphere above this height should, as a whole, gain as much heat by absorption as it lost by radiation ; the temper- ature of the convective region at this height then gave the temperature of the isothermal re'ion. It appeared that a satisfactory balance was obtained if the point of division was taken at a height given by p=+p,, where p is the pressure at any height, p, the ground-pressure. It appeared further that there was very nearly a balance of radiation in the upper * Gott. Nach. 1906, p. 41. + Annals Astrophys. Obs. Smithson. Inst. ii. p, 174 (1908). ‘The Sun’ (Appleton, New York, 1912), p. 323. { Astrophys. Journ. vol, xxix. p- 26 (1909). Phil. Mag. Ser. 6. Vol. 44. No. 263. Nov..1922. a) 874. Mr. B. A. Milne on Radiative Equilibrium : layer of the convective region extending from p=4p, to p=ip,, from which Gold deduced that in this layer the convection would be small. It is the object of this paper to point out a certain difficulty in Gold’s work, and to consider an idealized problem which is suggested by it in the absorption of radiation by an atmo- sphere subject to insolation. Since the paper was first written, the author has become aware of a paper by Emden* which anticipates portions of it. Emden criticized Gold’s theory on certain points, and investi- gated the general theory of the radiative equilibrium of an atmosphere by a method similar to that of the present paper. Where necessary, the paper has been recast to take account of Emden’s work. 3 § 2. Criticism of Gold’s solution.—One of the most in- teresting points in Gold’s discussion is his isolation and explicit formulation of the condition for a convective atmo- sphere. In such an atmosphere, transfers of energy are being effected both by radiation and by convection, and across any plane there will be a net radiative flux and a convective flux. Now convection can only transfer heat upwards, not downwards. But assuming a steady state, the upward convective flux plus the net radiative flux must be equal to the downward solar flux. Hence the net radiative flux (as due to the earth and atmosphere together) must be less than the downward solar flux. But the downward solar flux at any point cannot exceed its value at the boundary ; and at the boundary the downward solar flux must be equal to the outward flux due to the earth and atmosphere. Hence another form of the condition is : the net outward flux across any plane must be less than its value at the boundary. Again, the upper Jayers must be gaining more heat by convection from below than they are losing to layers above. Hence, for a steady state, emission of radiation must exceed absorption in the upper layers (for emission must equal absorption plus net gain by ecnvection). Whenever these inequalities become equalities, radiative equilibrium holds ; if they become re- versed the state cannot be a steady one, for it would involve convection downwards fF. * “Uber Strahlungsgleichgewicht und atmosphiarische Strahlung : ein Beitrag zur Theorie der oberen Inversion,” Sitz. d. K. Akad. Wiss. zu Miinchen, 1915, pp. 55-142. + Gold’s conditions have been applied by the writer to stellar atmospheres in a paper recently communicated to the Royal Society. the Insolation of an Atmosphere. 875 Now Gold applied these conditions in various ways to show under what circumstances a convective atmosphere can or einnot exist: e.g., he showed that a convective atmosphere cannot extend indefinitely, yet must extend above p= p1. But he did not point oat that his final solution was incon- sistent with these conditions. We shall show that although on the assumptions made the layer (47, 0) is neither gaining nor losing heat as a whole, yet its upper portions are emitting more than they are absor bing, and its lower portions absor bing more than they are emitting ; consequently the upper layers must cool and sink, the lower ones warm and rise, convection will occur, and the state of isothermal equilibrium must be destroyed. Further, although the layer (4), +p ;) satishes the conditions for convective > equilibrium as a whele, emission exceeding absorption, in the upper portions absor ption exceeds emission, so that a steady convective state in this region is not possible ; ; the smallness of the excess of emission cver absorption oe the whole layer, attributed by Gold to the slightness of convection required, is merely the result of the excess in the lower portions being balanced by the deficiency in the upper ones. Actually we can prove a more precise result than this, under very general conditions. We shall show that the excess of absorption over emission at the base of Gold’s isothermal layer, per unit optical mass, is numerically equal to the excess of emission over absorption at the top, whatever the temperature distribution in the convective layer and whatever the law connecting the coefficient of absorption with height. To do this we shall employ the approximate form of the equations of transfer of radiant energy. It may be inentioned here that though Gold uses the exact formule (involving #2 functions) which take full account of the spherical divergence of the radiation, his results can be obtained more simply to the same degree of precision by using the approximate formule and by making free use of the optical thickness and the net flux of radiation. The quite small errors of the approximate formule are swallowed up in the uncertainty of the numerical data that have finally to be employed. The uncertainty arises in the final trans- lation of the optical thicknesses into actual thicknesses; but, as in other cases of radiative equilibrium, many of the results hold in a form independent of the numerical values of the absorption coefficients. Let 7 be the optical thickness measured inwards from the aél2 876 Mr. E. A. Milne on Radiative Equilibrium : outer limit of the atmosphere ; if pis the density at height h, k(h) the mass-absorption coefficient, then ze) = ( k(h) pdh. oh Let I(r) be the intensity of radiation at 7 in a direction 0 with the outward vertical, where 0<@< 47; and let I'(7) be the intensity at yw with the inward vertical, where O=0°72138, B, =1'4427S, and | Se be eye be ee 72 Thus the values of T, and T, in terms of ‘7, come out about 1 per cent. smaller than on the previous approximation. The relation T.4=2T,¢ still holds. ‘The change is so trifling that we shall not attempt to obtain further approximations, which can be sought by using the integral equation. We shall 884 Mr. E. A. Milne on Radiative Equilibrium : content ourselves with observing that in the exact solution the differential coefficient B'(r) has a singularity at r=0, becoming infinite * like log rv. This is easily proved. § 5. Hatension to non-grey absorption.—Let us now suppose that the material has a different coefficient of absorption for the incident radiation, say equal to n times that for its own low temperature radiation ; » will usually be a small fraction. The inward solar intensity at 7 is now Se~”. Hence in the flux equation, (19), e~7 must be replaced by e-”", and in the equation of radiative equilibrium, (18), Se-7 must be replaced by nSe-"7. Proceeding as before, we find that I+ tn nr Bit) =8 fi (—dn)\enr"| i B, = S++2", B, = 38(1 +4), Tot yt Tot SS ban 1 le As n->0, T,->», To*>3T;', and the temperature dis- tribution tends to B(r) = S8($47). The limiting case is, in fact, the Schwarzschild case for a constant net flux 7F. Notice that TA= 2T)4/n. ~§ 6. Extension to oblique incident radiation. Next suppose that the external radiation is incident at an angle « with the normal. If we preserve the same intrinsic intensity, the amount incident per unit area is now S cosa and the amount crossing unit area at depth + 1s Sicosae—™ 5°¢*. |) Wemeamr yy obtain the solution by putting S cosa for S and nseca for n in the foregoing formule. We find B(r) = 8 tS [cos «—(cos «—4n)e—"* 2], (28"") B,, = Scosa(cosa+4n)/n, By, = 48(cosa+3n), To: Ty4: Tyt = cosa($+n-1 cosa): cosa: $(cosa+4n) = L4n-lcosa:1:4(1+inseca). . (29!) Notice that T,4=2 cos aT)4/n. * Cf. Monthly Notices, lxxxi. p. 3867 (1921). the Insolation of an Atmosphere. 585 §7. These formule offer several points of interest. As a increases from 0 to 47 and cos a—>0, |’) tends to a definite non-zero limit, although T, tends to zero; Ty steadily de- creases as a increases, the limit being given by oTy'=47n8 ; T.. tends to zero. It appears, then, that for sufficiently oblique incidence the boundary is warmer than the interior. Consider now the temperature distribution given by (28"). When cose=3n, B(r) is constant and equal to gS or S cos a> and the state is isothermal everywhere ; and when cosa J _ tenia eee TET 6 tka eee Pe T (31) When this averaging is taken into account, the approxi- mately isothermal state (T,.=1)) is found to oceur for cosk=n/¥ 2; for this value of X, Tyi/T'= 4. +40 V2) = 1-055, which is sufficiently near unity. The general run of the change of temperature distribution with latitude has the same features as before. § 9. Comparison with imden.—The formal problems dis- cussed by Emden in the paper already mentioned and his method of solution are very similar to those discussed above, except that he takes the material to be bounded below by a black surface. Emden considers the radiative equilibrium of an atmosphere subject to external solar radiation in two cases: (1) the case of “grey radiation,’ by which he means the case in which the mean coefficient of absorption for the solar radiation is equal to that for the atinospheric radiation; this is the case n=I1 above; (2) the case in which the radiation spectrum can be divided into two ranges which have different mean coefficients of absorption, the solar radiation being entirely confined to one of them; this is practically our general case in which v is not unity, In each case he considers the solar radiation to be “ gleich- miissig verteilt,” i.e. not as being confined to a parallel beam, but as uniformly distributed over the solid angle 27 ; consequently he does not consider the variation of the state of equilibrium with latitude. The two main results to which he draws attention are: in case (1) the whole atmo- sphere must be isothermal, at a temperature equal to the “effective”? temperature T, calculated from the incident radiation with allowance for the albedo (see p. 873 above) ; in case (2) the state is not isothermal and the boundary temperature Ty) is connected with the effective temperature c the Insolation of an Atmosphere. 887 T, by the relation re eit vei) ha) | ay) ) eG eee) (82) where /; and i, are the coefficients of absorption for the solar and terrestrial radiations. Both these results are in apparent contradiction with those obtained in this paper. The source of the discrepancies is in each case Emden’s assumption that the incident radiation may be taken to be diffuse. The way this occurs is as follows:—The mean coefficient of absorption for diffuse radiation incident on a thin layer of material is approxi- mately twice the coefficient of absorption for a parallel beam incident normally, 7. e. twice the coefficient of ab- sorption as ordinarily defined. This fact allows us to approximate to the equations of transfer (equations (1) and (2) above) by replacing them by the ‘equations of linear flow”; in equations (5) and (6) we have explicitly adopted a new optical thickness ¢ equai to twice the optical thickness t obtained directly from the ordinary coefficient of absorption ; in equations (24) and (25) we have retained the optical thickness t and simply replaced the factors cos 0 and cosy by the value 3; the result is the same as if all the diffuse radiation were supposed to be confined to beams at an angle of incidence of 60° with the planes of stratifi- eation. Emden approximates in the same way as we have done, but since he takes the solar radiation to be diffuse he is adopting for this also a coefficient of absorption twice the value for a permanent beam. His results may therefore be expected to agree with ours if in ours we put cosa=}3, a=00°; and this in fact they do. But they lose part of their significance. His result for case (1) is of course true for diffuse radiation; indeed it is obvious thermo- dynamically, without proof, that material exposed to iso- tropic incident radiation will, if in radiative equilibrium, take up a temperature equal to that of the radiation: the case is practically that of a black body enclosure. But our results show that if the incident radiation occurs as a parallel beam—as, in fact, solar radiation does—-then the isothermal state is merely the particular distribution of temperature that happens to correspond to an angle of incidence of 60°. Further, Emden’s result does not suggest another of our results—that when n¥1 there also exists an isothermal state of equilibrium: namely, for cos a=4n for a fixed parallel beam, and for cosX=n/ V2 when rotation is taken into account. Hmden’s formula (32) 888 Mr. E. A. Milne on Radiative Equilibrium : above should be compared with our formula for a fixed beam ‘Incident at @ (from (29’’)), To = T,[3(.+4nseca)]?, . . , . (BB) where n=hk,/k,; and with the rotational mean formula fro1 ) CCIE, Ta aan sco an Hmden’s formula differs but little from the latter when 7=0, as is to be expected. Emden does not obtain the integral equation for the temperature distribution. For the sake of completeness it seems worth putting on record the integral equation for the general case involving n and «. It is deduced in the same way as (22) :— OMG esa) Cir) daemon af - e~vdy +| AGED esi, eo § 10. Liffect of an internal boundary.—We shall next consider the case in which the material is bounded internally by a black surface at t=7, insiead of extending to infinity. It has already been mentioned that as the formule only involve the optical thickness 7, we may deduce the results for this case by supposing that immediately beneath t=7, the density suddenly increases indefinitely. The temperature distribution above 7, is unaltered. It might at first be supposed that the black surface would assume a temperature equal to T,,, but this is not so. For the infinite density gradient we have postulated at the level +, implies an infinite radiation gradient there, and (unless we are prepared to accept the existence of an infinite temperature gradient at the black surface) the surface will take up a temperature intermediate between T(7,), the temperature of the material in contact with the surface, and |’,. This temperature, say ‘l,, is easily cal-. culated. For since the surface must re-radiate all the radiation falling on it, we shall have oT,4=7B,, where B, is given by . By = UGS cos ae "Roo aes, e the Insolation of an Atmosphere. 889 From the equations eee eee cd ko LOY Pesaro cone rrrese ii) i CO eunad I’ = B—438e-"7 *(cos «+ 4n). Hence Bs: = B(x) +38(cose—injem sera, | | (37) If cos a>4$n, which will usually be the case in applications, B, is greater than B(7,). Thus the temperature of the surface exceeds that of the material (say air} in contact with it. Hence convection currents would be set up, and the state of radiative equilibrium would be destroyed. his is a simple way of demonstrating the impossibility of the existence of a state of radiative equilibrium throughout the entire atmosphere. § 11. The “greenhouse” efect.—Inserting in (27) the value of B(7,) from (28'’), we have B, = 8 cos a[ (cos a+ 5n)— (cos a—3n)e-™71 82] In, (38) Now if the black surface were exposed to the direct in- solation 7S cose, without the intervention of an atmosphere, it would take up a temperature T,’ given by ol, */7 = B. = Scosa. Hence Me (cos a—4n) (1 —e7771°¢¢2) pa = 1+ : (39) Thus, when cos «>4n, the surface is maintained at a temperature higher than it would be in the absence of au atmosphere. The ratio T,*/T,’* increases as n decreases, the limit as n—>0 being 1+7;. The case of diffuse incident radiation is roughly given by putting cos#=4, and then the condition is n<1, 7%. e. that the atmosphere or “ protecting layer ’ must be more transparent to the incident radiation than to the radiation returned. This is the radiation part of the ‘“‘ greenhouse” or “ heat-trap” effect, which is some- times the subject of fallacious statements ; 1t must of course be distinguisbed from that part of the effect which is due to the prevention of convection. . § 12. Extension to a partially convective atmosphere—We will now generalize the problem a little further. Suppose that we have a state of affairs in which the material above a Phil. Mag. Ser. 6. Vol. 44. No. 263. Nov, 1922. 3 M 890 Mr. E. A. Milne on Radiative Equilibrium : given level t=7, is in strict radiative equilibrium, that below the given level merely in radiative equilibrium as a whole ; below 7=7, the temperature distribution may be of any form (with or without a lower bounding surface) subject only to the condition that the whole system below 7, radiates out- wards as much as it absorbs; in general, convection of heat will be required in the region below 7=7; in order to maintain a steady state. Then it is easily seen that the temperature distribution above 7, is exactly the same as if the lower region — were in radiative equilibrium in the strict sense ; for the upward intensity at 72, namely I(7,), is the same in the two cases. Hence the temperature distribution we have already found applies to the region above t,. The importance of this point from the point of view of applications to the earth’s troposphere and stratosphere is evident *. § 13. Zhe boundary between troposphere and stratosphere.— It is convenient to denote the regions below and above the _level 7, in our ideal problem by the words “ troposphere” and ‘stratosphere ”’ respectively, without implying any reference to these actual regions in the earth’s atmosphere. Then §12 shows that under the conditions there stated a stratosphere cannot be isothermal unless its optical thick- ness is zero or cosa=4n. If the optical thickness is not zero and cos « >4n, the lower parts of the stratosphere must be warmer than the upper. ‘This agrees with § 2, where it was found that Gold’s stratosphere is warming up at the. base. We are now in a position to frame in a precise manner the problem of where the division between troposphere and stratosphere should occur, in the ideal case. Let us suppose that there is a certain distribution of temperature which the processes of convection tend to set up throughout the whole atmosphere. Let the corresponding black body radiation- function be expressed as a function of optical depth, say B.(7). This temperature distribution together with the lower boundary surface implies a definite upward intensity of radiation at any point 7, say I,(7), which is determinate and caleulable when B.(7) is given. Let tT. denote the optical depth of the surface of separation between troposphere and stratosphere which it is required to determine. Then * The points which are the subject of §§10, 11, 12 are substantially made by Emden, in the form appropriate to diffuse radiation. But Kmden’s analysis is in parts a little complicated by his introducing unnecessarily early into the investigation an empirical expression for the water-vapour in the earth’s atmosphere as a function of height. (the the Insolation of an Atmosphere. Sor below 7, the temperature is given by B,(r) ; above 7 it is given by the function B(r) given by formula (28"). At 7, the upward intensity of radiation is that appropriate to the state of radiative equilibrium ; it is the value I(7,) deducible from (26’’) and (27'’), I(1s) = B(rs) +38 (cosa—4nye-rs we, Flence v2 is the root of the equation cee a Ugreise tin O20 0°83 hat | 050 1°88 vi) OOL +.) G:05° 0-28 | 0:02 | 0-10 0:69 | 0:03 | 0-23 | 1:38 |-0:10 | 0-45 | 277 | 0:20 | 088 | Sodium Chloride. | 063 | O11 | 0:38° | 1°36 0:23 0°80 P Bibl 0-43 1°45 | 410 | 070 | 2-40. Potassium Chloride. 1°56. | O21 0-71° 310 | 0°42 1:40 5 60 | O75 2°53 { pS Potassium Carbonate. Observed| Normal De- pression.) pression, 0-02° 0-04 0:09 0-19 0°37 0-21° 0-43 0°80 1:30 0°39° 0-78 1°39 Increase. Surface, Tension 74°23 74:57 7452 7457 7486 75°04 75°22 74°40* Te. 75°06 75°69 74°32 74°50 74:80 75°30 74:19 74°32 74:50 74:83 THxDT T415 74:18 14:28 74°48 74°89 74°32 74:54 74°91 75°40 74°40 74:70 75°10 | | ‘Increase. 0:13 0:47 0:42 0-47 0-76 0-94 1°12 0°30 0°64 0°96 1:59 0:22 0°40 0:70 1:20 Ratio. 0-61 0°62 0-64 | 0°65 | 0-73 0°75 O71 0°80 0°69 0°62 0°57 0-64 0-60 0:63 0:69 0:69 063 0°64 0°63 + 0°65 | | | 0-64 | | O-75 | 0-79 | 068 0-64 0°85 0°88 0:82 0 86 0-80 | O90 | | 1:16 903 eg ! | Surface | | Tension | | x 0°64. | | | * These figures were calculated from the two sets of determinations at different temperatures. 904 Mr. R. A. Mallet on the Failure of intrinsic pressure, and Walden * has calculated the value of — the ratio from determinations made at the boiling-point. If surface tension is really proportional to intrinsic pressure, the same ratio should hold at ordinary temperatures, nor is there anything in Walden’s calculations inconsistent with this. Accordingly we may write OII=75'36c, dII being the increment of intrinsic pressure due to the non- ageregated solute particles and 6c the corresponding in- | crement of surface tension in dynes per centimetre. Since 1000 atmospheres increase of pressure depress the freezing- — point of water 8°:5, the depression, 6D, corresponding to the increase of internal pressure dII would be 60 X75°3 x 875 a a de : 1000 = 0°64 60%. S00 0) io The correspondence between the observed increases in the depression and the calculated values of 0°646c, shown in the last column of the table is remarkable. It appears, therefore, that the observed increases in the depression of the freezing-point of aqueous solutions of salts above that due to a normal solute, are caused by the enhanced intrinsic pressures resulting from the greater molecular fields of the solutes. For non-aggregated salts the increased depression is given by the equation (v.). Incidentally this may be regarded as an experimental verification of Walden’s relation. The Science Museum, South Kensington, London, 8S. W. 7. LXXVI. Onthe Failure of the Reciprocity Law in Photography. By R. A. Matuet, B.A.+ VNHE failure of the photographic plate with a silver bromide-gelatine emulsion to obey the ‘‘ Reciprocity Law” of Bunsen and Roscoe was first observed by Abney f. The first quantitative work on the subject was done by Schwarzschild §, who proposed as an empirical “ Law of * Zett. physik. Chem. vol. lxvi. p. 885 (1909). + Communicated by Prof. ‘T. R. Merton, F.R.S. t Proc. Roy. Soe. liv. p. 143 (1893). § Astrophysical Journ. xi. p. 89 (1900). the Reciprocity Law in Photography. 905 Blackening”’ the following relation : See De where 8 is the degree of blackening, I the intensity of the incident light, T the time of exposure, and k& and p are constants. The constant & is fully discussed by Plotnikow *, and depends on several factors, mainly on the thickness and composition of the film. The exponent p is known as the Schwarzschild constant. Its value was determined by Schwarzschild for Schleussner plates, which he exposed to the light of a normal benzene lamp at various distances for varying lengths of time, measuring the depth of blackening by comparison with a scale of blackness standards made by means of a Scheiner sensitometer. He found p=0 86. Other workers in this field are Becker and Werner 1, Leimbach {, Helmick § who used a Lemon spectrophoto- meter, Stark || who used a Kénig-Martin spectrophotemeter, and Plotnikow who used a Kruss polarization colorimeter. Apart from the fact that all these workers have obtained their results by methods depending on the comparison of light and shade by eye, and that their numerical results vary some- what, no one appears as yet to have suggested any physical meaning for the constant p. It is, in fact, uncertain whether it depends on the density, and whether it is constant for a given make of plate under all conditions. Stark has shown that it is subject to considerable variation over abnormal ranges of exposure, and he and others have shown that it varies somewhat with the wave-length of the light used. Schwarzschild (doc. cit.) used both light and heavy blackening, and found it to remain constant, but some further investigation seemed desirable. It was decided to use a method which did not depend on visual comparison of varying shades of blackness, and to this end a Goldberg wedge screen was introduced between the plate and the source of light. This was carried out as follows :— A brass plate, having in it a slit about three inches high and three millimetres wide, was let into a wooden hoard, having at the back a recess of such size that a quarter-plate wedge screen fitted in closely, in such a position that the slit was opposite the middle of the wedge. Immediately behind the wedge was placed the plate-holder, consisting of * Lehrbuch d. Photochem. p. 667. + Zeit. f. wiss. Phot. x. p. 382 (1907). t Zeit. f. wiss. Phot. x. p. 137 (1909). § Phys. Rev. xvii. 2, p. 142 (1921). | Ann. d. Phys. xxxv. 3, p. 461 (1911). Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3N 906 Mr. R. A. Mallet on the Failure of an ordinary single metal dark slide, so arranged that it could be slid across the slit and stopped in any of several positions by means of a small spring loaded plunger. The wooden ‘board was fixed to the bench in an upright position, and a well-fitting wooden shutter arranged to slide across the front of the slit. This shutter was worked by hand. The source of light wasa 36-watt 12-volt gas-filled motor headlamp bulb, connected in parallel with a voltmeter to a 12-volt accumu- lator battery. This battery was connected through a suitable resistance to the town mains so as to form acharging circuit. In the course of three months the voltmeter reading fell by | 0-2 volt, so that the drop during one experiment was negligible. The lamp was enclosed in a light-tight box with a square hole cut in the front, with the object of removing irregularities due to stray reflexion from tke walls of the room. The lght was used at a constant distance of about one metre from the plate, at which distance even illumination over the whole of the slit was obtained. All exposures were timed by a stop-watch, the error in moving the shutter being of the order of 1/10th of a second, which may be neglected, in view of the length of the exposures used. Jt is evident that with an arrangement of this kind, if a plate be exposed behind the wedge, a line will be photo- graphed on it, the length of which will be a function of the time of exposure and of the intensity of the incident light. Furthermore, if two lines are photographed side by side and the time of exposure varied while the intensity is kept constant, then Tyee Typ = Tyee TY, _ __ plt—th) and hence p= log Ty lee 8? where p is the wedge constant, T) and T, the times of exposure, ¢, and ¢, the lengths of the lines. Determination of the Wedge Constant. | The wedge constant was determined for the light used by exposing Paget “ Half-Tone”’ plates behind it to light of two different intensities, the time of exposure being kept constant. The intensities were in the ratio 1 : 25, the variation bein obtained by altering the distance between the light and the plate from one metre to five. Then, since Tp>=T,, Ter = Tie 9h, oe log I,—log I, and hence ty —ty the Reciprocity Law in Photography. 907 The following are the values obtained :— Exposure (mins.), ¢) (mm.). ¢, (mm.). 0. 30 33°10 14:79 0:0754 "32°96 14°65 00764 30 34°92 16°34 0:0740 25 34°87 15°80 0:0733 34°72 15°51 0:0728 2°5 36°72 18°64 0:0773 39°66 16°48 0:0729 25 34:08 16°23 00783 33°98 15°24 0:0746 2°5 35°54 17-74 0:0725 2°5 46°07 27°34 00746 In the case of the last of the above results the plate used was a Wratten ‘“ Instantaneous,” which is considerably faster than the Paget. The mean of the above is 00747, with a probable error of +0°0004. The mean obtained is somewhat higher than the value given by the manufacturers, Messrs. Ilford, in their calibration ot the wedge used, but this is accounted for by the fact that the wedge constant increases in the violet and ultra-violet. Toy & Ghosh*, who investigated this point, found that the value of p begins to rise at a wave-length of about 4500 A, -at which point the density of their wedge was 1:25. The density at 4000 A they found to be 1°6 and at 3000 A about 3:5. A usual method of calibration is to focus the (small) filament -of an electric lamp on various parts of the wedge, and to throw. the spot of hght passing through on toa white screen. Light froma similar source is passed through two Nicol prisms on to an adjacent portion of the same screen, and the two spots adjusted by means of the nicols to the same intensity. Since the normal photographic range extends further into ‘the ultra-violet than the visual range, the value for p given by this method will be lower than that obtained by the photographic method. Determination of the Schwarzschild Constant. Three types of plate were used, so chosen as to give a large ‘variation of properties. They were the Paget *‘ Half-Tone,”’ the Wratten “ Instantaneous,” and the Imperial ‘‘ Eclipse.” ‘The Paget “‘ Half-Tone” is a process plate, about one-third faster than wet collodion plates, and giving great contrast. ‘The Imperial “ Eclipse ” is one of the fastest plates obtainable, -and the Wratten “ Instantaneous” is intermediate between ‘the two, being a medium slow landscape plate. * Phil, Mae. Dec. 1920, p. 775. 3.N 2 908 Mr. R. A. Mallet on the Failure of The approximate: relative speeds of these plates were obtained by exposing a specimen of each behind the wedge to light of the same intensity for the same length of time, and measuring the lengths of the lines obtained as nearly as possible without further treatment of the plates after develop- ment. Then, at the end of each line, TS)? cme’ =e where S is the speed of the plate and e* is a constant. Since p is not identical for each plate, it is necessary to reduce all lines to a standard length, and take the ratio of the actual time to the time for that length as proportional to the speed. It ¢,, t,, etc. are the actual lengths of the lines, and ¢, the standard length, then _ loglh>p—k+>p log (ST) t= p aon = wet eagles (ST). : p al Mf ee = log ks, OS mr T, ae 0) == Si Oe perp i zal = — ( Gilat + Te | to—t,). 08 y Di 2) es Ds (t2—to) Hor dametiplatesy au press Mik O°0864, t==2i53 » Wratten plates ......... p/p=0'0884, t=38-0 », Lmperial plates ......... p/p =0°0884, ¢=46°7. t) was taken as 35, which is approximately the mean value of t. From these figures the relative speeds of the plates are roughly :—Paget 1, Wratten 29, Imperial 165. The Schwarzschild constant was determined by photo- graphing lines on the plate with long and short exposures alternately, the intensity of the light being kept constant throughout, and the duration of the long exposures being usually one hundred times that of the short. The actual times of exposure are given in the table below. In the case of the Wratten and Imperial plates, the intensity of the incident light was reduced to a convenient extent by inter- posing one or two neutral screens between the source of light and the wedge, the screen being let into the front of the box in which the light was enclosed at a distance of about 5 cm. from the filament. This was necessary, as, owing to the sensitivity of these plates, the exposures would otherwise have had to be made too short to admit the Reciprocity Law in Photography. 909 of accuracy in the timing. The plates were fully developed in hydroquinone and caustic soda developer, and “cut” by a few seconds’ immersion in potassium ferricyanide to remove the slight chemical fog and to give a more easily measurable end to the line. After this treatment, the Paget, and in one or two instances the Wratten, plates could be measured directly, but in other cases the plates had to be printed onto Paget’s, and one, or sometimes two, further transparencies made before it was possible to measure them. The plates from which the wedge constant was determined were treated in the same manner. All measurements were made on a Hilger travelling microscope, and are accurate to ‘1 mm., and in some cases to °01_mm. : As is well known, irregularities occur near the edges of a plate. For this reason, only two long and two short lines were made on each plate, and thus only the central portions were used. It will be seen that the results for. the same type of plate differ appreciably, but this is to be accounted for by variations in the thickness of the film on the plate, and the errors arising in this manner are eliminated by taking the mean of a considerable number of determinations on different plates. Paget Plates. Wratten Plates, Tmperial Plates. Exposure E ! E : Gaamdsy. D. xposure. p. « :. Exposure. Pp. 32 & 15976 0-830 10 &1020 0-874 20 & 2000 0°848 i0 & 1000 0°840 0°845 32 & 15976 — 0°833 0-900 10&1000 0867 10&1000 0-868 10&1020. O861 | 0°880 60 & 6000 0-867 0879 10& 1008: > 0857 —- 10% 1000" © 0's 61 0°798 0:897 60 & 6000 0:856 0°873 10&1000 0:867 10&1000 0854 - 0°834 0-861 45 & 4500 0-792 0 817 10 & 1000 0°831 10 & 1000 0°765 0-840 0°830 . 45 & 4560 0°962 0-902 5 & 550 0-878 10&1000 0847 0°837 45 & 4500 0°861 10&1000 0-834 0-871 0815 10&1000 0°808 0°865 0°865 45 & 4500 0°847 _ 10&1000 ~ 0829 0°846 30 & 2000 0-869 30 & 2000 0°885 0°868 910 Mr. R. A. Mallet on the Failure of From the foregoing are obtained the following mean values of p :— Paget “* Halt-Tone”’ plates v.40... 0:865+ 0:005 _ Wratten ‘ Instantaneous” plates. 0°846+0°005 Imperial ‘‘ Eclipse” plates ......... 0°3846 + 0-005 The probable error, which is certainly not due to errors in measurement of the lengths of the lines, was calculated i in the usual manner. It is noteworthy that the mean values for the fairly slow Wratten plates and for the very fast Imperial plates agree exactly. Becker and Werner (loc. cit.) have stated that the value of p tends to increase ae sensitivity, but this does not appear to be confirmed by the present investigation. The question then arose as to whether p was really a constant for any one type of plate for any variations of intensity and duration of exposure within the normal range (great over-exposure, 2. e. solarization, being left out of the question), or whether it was dependent on the intensity or on the time of exposure, or on both. Hvidently, if p is constant for all conditions of time and intensity, the gradation of two lines on the plate of about equal length, but made under differing conditions, will be the same. To ascertain this, two plates were prepared, having on them lines of approximately the same length, but in which the intensity and the duration of exposure were so altered that IT?=const. Six lines were made on the two plates. The intensity of the incident light was as 1 in the case of three lines to 6°25 in the other three, and the time of exposure adjusted so anne IT’=8 in two cases and =12 in the other four. The plates used were Paget “‘ Half-Tone.” They were fully developed but not “cut,” and the relative blackening at different points of the lines was compared by means of a photometer, involving the use of a photoelectric cell and a string galvanometer which had been designed by Mr. G. M. B. Dobson, to whom I am greatly indebted for the loan of this instrument and for his assistance andadvice. Owing to the limited range of the galvanometer it was possible to measure the densities of the lines only over a range of 10 to 14 mm. The distances from the ends of the lines were plotted against the actual galvanometer readings, which have been used as an arbitrary scale of blackness. It will be seen from the figure that the gradations of the lines are identical, — the Reciprocity Law in Photography. 911 despite the alterations of intensity and duration of exposure. The points represented by +A etc. refer to the different plates measured, and the values plotted are over the same range of density so as to show the similarity in the shape of the curves. From this it is evident that the Schwarzschild constant has a definite value for each type of plate, and that this value remains constant over a range of density extending from zero to approximately unity, since the density of the wedge used is about 0°075 per mm. and the portion of the lines considered in the curves given is about 14 mm. long. Fig. 1. 600 uw °o oO Calvarometer Readings. b ° ° Millimetres. It is further evident from a consideration of these curves that the value of the Schwarzschild constant is the same at the “threshold,” 2. e. at the point where the plate is just darkened, as at greater densities. Owing to the “cutting ” of the plates used in determining the constant, the value given by them was not that at the ‘‘threshold,” but the measurements made with the photvelectric cell extend to the extreme ends of the lines, and the curves coincide perfectly at this point also. 912 Messrs. Roberts, Smith, and Richardson on It is evident that the Schwarzschild constant has a very definite physical significance, but what this may be it is not possible to say. From the values obtained with the three types of plate it is evident that it is not greatly dependent on either the size of the grain or on whatever factors ulti- mately determine the sensitivity of a plate to light, but a fuller knowledge of the composition of the silver halide- gelatine emulsion and of its physical properties would seem to be necessary before a complete explanation can be arrived at. This investigation was undertaken at the suggestion of Prof. T. R. Merton, F.R.S., to whom I am deeply indebted both for the loan of almost all the apparatus used, and for much valuable help in the course of the investigation. Thanks are also due to Mr. C. H. Bosanquet for several valuable suggestions. Balliol College Laboratory, Oxford. LXXVII. Magnetic Rotatory Dispersion of certain Paramag- netic Solutions. By R. W. Roperts, M.Sc., J. H. SMirs, M.Sc., and 8. 8. Ricnarpson, D.Sc., A.R.C.Sc.* NE of the anomalies met with in the examination of the Faraday effect is that, whereas the salts of iron in solution give rise to a negative rotation, those of the strongly magnetic elements, nickel and cobalt, produce a rotation of the plane of polarization in the opposite sense—that is, in the direction of the rotation produced by the great majority of compounds both inorganic and organic. In the case of cobalt sulphate the rotation, though positive, is almost zero. It is noteworthy, however, that the measurements on which such statements are based have been made only with refer- ence to the D line or at most a few lines in the visible spectrum, and some years ago it occurred to one of us that further information might be obtained by examining the course of the rotatory dispersion in the ultraviolet. Hxperi- ments were carried out in 1916 on CoSQ,, and the spectrum photographs showed that the rotation of this salt becomes strongly negative in the ultraviolet. The principal absorp- tion-band lies on the borders of the visible and ultraviolet, and the rotation, which has a small positive value up to the edge of the band in the visible spectrum, becomes negative where the spectrum reappears, and remains negative through- ‘out the ultraviolet up to the point where general absorption * Communicated by Prof, L. R. Wilberforce, M.A. Magnetic Rotatory Dispersion of Paramagnetic Solutions. 913 cuts off the spectrum completely, which occurs in the region of 3000 A.U. The result with cobalt indicated that a systematic ex- amination of the the rotation produced by paramagnetic substances in the ultraviolet would be of interest, and the present communication refers to the results obtained with the sulphates (fig. 1), chlorides (fig. 2), and acetates (fig. 3) of nickel and cobalt, also ferrous sulphate, in aqueous solution™. hea ration IN RADIANS X10 Su LPHATES -2 » A x lofu 4) The method employed has already been described in connexion with the rotation in certain organic compoundsf. The solution was contained in a short tube (1°135 em. long) placed between the poles of the electromagnet. The rotation obtained for water showed that a reversal of the current (7 amperes) produced a change of magnetic potential 30380 Ci.-2auss. In the accompanying graphs dispersion curves are given * The results for nickel and cobalt sulphates were confirmed in this Laboratory in 1918 by the Rev. W. D. Ross, M.Sc. + 8. S. Richardson, Phil. Mag. vol. xxxi. p, 282. 8s go 95 * 100 [ots “uo 914 Messrs. Roberts, Smith, and Richardson on Fig. 2. | Ve CHiorI DES \\ fROoTATION IN RADIANS x 1072 e, es 5 RADIANS x /0 od ® \ ‘\ a Ro TATION G fo} 1S 80 85 qo qs 100 (os ula Magnetic Rotatory Dispersion of Paramagnetic Solutions. 915 for the solutions and for pure water*. When the water of crystallization of the salt is taken into account, the mass of water per c.c. is practically the same in the solution as it is in the case of water alone; the difference in the ordinates therefore represents the rotation produced by the salt alone. As small differences of rotation were to be measured it was necessary to work with fairly concentrated solutions, and the graphs refer to the following concentrations :— Mickel, Chlomdep os 30). tasaacs « ‘216 molar. Cobalt BS ge ont Fe ee he Sp GED es, Nickel Sulphate. gi< <% ozsejes ‘624 =, (ne ae Se Pare eke Ee rr “B04. 55 Ferrous _,, 5.9 sells tr cn EU Se Niekel Acetate (sid. 6.8. evs “664. * Cobalt BM a a ae a act "20K - J. The results indicate that the cobalt atom, like the iron atom, is capable of producing a negative rotation, but of smaller numerical value. ‘The disappearance of this on the low-frequency side of the band may be attributable to a preponderating positive effect of the (SO,), (C,H;0,), Cl ions. This view is supported by a fact that Sel a highly dispersive element (e.g., chlorine) is present the residual negative effect in the ultraviolet is smaller, but the point requires further investigation. In the case of nickel the rotation remains positive pageae ee: the range mea- sured, but does not undergo the large increase in the ultraviolet which is observed with diamagnetic substances. It is possible therefore that the nickel atom, whilst not able to overbalance the positive effect of the other atoms in the salt, still exerts sufficient depression nearly to neutralize the dispersion. From this point of view, our results indicate that in respect of paramagnetism the cobalt atom occupies a position intermediate between those of iron and nickel. _ The experiments are to be continued, and we hope to publish results for other paramagneties shortly. We beg to tender our thanks to Prof. Wilberforce for the facilities and apparatus placed at our disposal and to Dr. Smeath Thomas who kindly prepared the ferrous sulphate solution. The George Holt Physics Laboratory, ‘University of Liverpool. * To obtain the rotation of the salts in radians per cm.-gauss the ovaph-readings must be multiplied by 6°616x10-’. —— is O16. 4 LXXVIII. Colour- Vision Theories in Relation to Colour- Blindness. By F. W. Hpriper-Green, U.B.E., M_D., F.R.C.S., Special Examiner and Adviser to the Board of Trade on Colour Vision and Hyesight * ee importance of colour-blindness as a key to any colour-vision theory does not seem to be sufficiently recognized, though the fact was well known to Helmholtz f, who showed that Hering’s theory explained the facts of colour mixing quite as well'as his own, and stated: ‘‘ As far as I see, there is no: other means of deciding on the elementary colour sensations than the examination of the colour-blind.” This cannot ‘be too widely known, because any other method assumes that the three-sensation ‘theory i is correct, and is useless when ‘this theory is denied. As has been shown by Houstoun {, my non-elemental theory explains the facts of colour mixing quite as well as either of the above mentioned. Recently I have examined about 200 cases of colour-blindness by colour-mixing methods as well as my own, in order to ascertain certain facts of crucial importance in deciding between an elemental and a non-elementual _ theory. Relation of Luminosity to Colour. If the sensation of white were compounded of the addition of three elementary processes and one of these processes were subtracted, the position of the apex of the luminosity curve would not be the game in the colour-blind as in the normal, It is, however, well known that there are numerous dichromics rhe have a luminosity curve aL to the normal. A case of colour-blindness regarded from the point of view of a three-sensation theory may, for instance, be one-half red-blind ; the composition of the theoretical white will then be 4R+1G41V. As far as luminosity is concerned, this white may be compared with light of various wave-lengths by the colour-blind subject, just as normal white is by the normal sighted. Now, as the apex of the luminosity curve depends upon the point where the aggregate stimulation of the three theoretical sensations is greatest in terms. of luminosity, this apex will be displaced towards the point of maximum stimulation of the other sensation, namely green, the luminosity of the blue being so low as to be negligible. Numerous cases can, however, be found in which the apex of * Communicated by the Author. + Physiologische Optik, 2nd edition, p. 377. t Phil. Mag. vol. xxxviil. p. 402 (1919). Colour- Vision in Relation to Colour-Blindness. 917 the luminosity curve is at the same point as the normal. c= kp ey 2\1/3 .2\ 1/3 2\ 1/8 TOI, _ ) U. — — ) V, w= ae ) W, cy Cy ey / (B22 ps (=== )n. It is readily found that the substitutions eliminate the parameters satisfactorily if cv/k is a constant and equal to unity. For liquids ecv/k may have very large values (glycerine 8000, etc., see later), but for gases, as indicated by the Kinetic Theory, it is constant and approximately 3.0 2 924 Mr. A. H. Davis on Natural equal to unity. For gases, therefore, the above substitu- tions should be fairly satisfactory on this ground. There is an alternative condition under which the substitutions are satisfactory, even without cv/k being a constant. The condition is that the accelerations wu’, v’, w’ of the particle shall be negligible compared with vV/7u, etc., which would appear to be justifiable for very viscous fluids. It implies that on coming into the region of the hot body a particle of the fluid almost immediately takes up its final velocity and suffers but little subsequent acceleration. Consequently, the above substitutions appear satisfactory, and the differential equations (3) and (4) take the following form.) (0° V \Vmay ibe retained if viet but otherwise they must be omitted since wu’, v’, and w’ are neglected *). 0oU ov OW pe +2T oT or \ aft ay) OE a” Vasataa tom Se US see) ow Se Oe ten ae | Ol gy (ON O eae By 7 te tat * ae) ol | ew ow Pee a ee W'+ OF =F On” D) + 0’ | where (U’, V'7,W,T) = po, ee al) | + VRC VW) 4 yy 90 MD) | On * IT am indebted to Mr. W. G. Bickley, M.Sc., for the following notes :— (a) If ey/k is not ae to unity, equations 8 are mathematically ele if we write a ow for U’, etc. Evidently, if cv/k is large, the term ie of is correspondingly small and may be omitted. Retaining it, however, the solution of (8) becomes T a P : ; Q? Caan (ee a B = definite functions of | (nas Bh The experimental curve shows that the cccurrence of cy/k in these functions is in such a manner that large variations in cy/k have imperceptible effects. (6) Equation (5), and the resulting one (8), would be more general if the O(24,0,@,7) cy term were introduced on the left-hand side. This would include unsteady motion, but would in no way affect the changes of variables. (8) Convective Cooling in Fluids. 925 Also, let us put the equation of the solid in the form A pee: ae (“ee )" | =o. es) Thus, if the coefficient (@ gac/kv) changes, this amounts to considering, instead of the actual solid, similar bodies having linear dimensions inversely proportional to this coefficient. Then the direction-cosines 1, m,n, of the normal will remain the same at corresponding points, and the boundary con- ditions become At the surface U=V=W=0 and T=1, At the distance 4/£+7?+ ¢? infinite (II, U, V, W,T)= a - (10) The system of equations (8) and (10) determine (U, V, W, T, II) as the functions of &7& and substituting in the integrals for the eight new variables their equivalents as given by (7), we have five relations of the form me (u,', w) EP , 1 0’ aye ee ae = definite functions of | Cy pP Cc ‘ ; (11) \ Ogac\\? Ogac\' Ogac\? ee 4 ce) 2 oa ey The flux of heat furnished in unit time by unit area of such a body, equal to that which the contiguous liquid layer communicates to the interior the fluid, is given by k Ge +m = 7 t” £7), Introducing the new set we have then =k(é gaciiv) (1 =~ +m oe +n g) vRELZ) At corresponding points of the eae f (&, 0, )=0 limiting the bodies considered, the direction-cosines 1,m,n and the derivatives sto 2) have the same values respec- y) n; tively ; so the ee. coefficient is a function of the shape and orientation of the bodies only. Thus the result may be stated in the following form. For a family of similar bodies similarly oriented, and haying linear dimensions L given by Lice (6gae/kv)— 8, i.e. (L?@gac/kv) =const., 926 Mr. A. H. Davis on Natural the heat loss per unit area from corresponding points is given by 3 NeackeOgacky) ?, see and this will also be true if 2 represents mean heat loss for the whole model. So for bodies of this shape and orientation we have : h=ki(Ggacikv)? fArdgaciky)) .:). eae which may be written hlyjkO=Ve0gacky).. - 2 ee This equation is the simplified form of equation (1) it was desired to establish, and it has been put to the test of experiments in a later part of this paper. It is desirable to notice here one point in connexion with it. : For a series of fluids for which cv/k is constant, the equation may readily be shown to agree with that obtained by Boussinesq for inviscid fluids ; 2. e. Boussinesq’s grouping of variables for invisicid fluids is satisfactory for viscid fluids for which cv/k is constant. This equivalent grouping is given by omitting (cv/k) from formula (1). Part [I].—ExPpERIMENTAL. Formule. For long horizontal wires of diameter ‘d’ it may readily be shown that formule (1) and (2) may be rewritten as follows in terms of the heat loss H per unit length of wire per degree temperature excess : emileriaieiae (16) H/k= F(d?8gac/kv). When cv/k is a constant, the equations are identical in form and, consequently, evidence for diatomic gases already shown elsewhere to be in agreement with one of these expressions is necessarily in agreement with the other. If cv/k is not constant, the second equation is a special case of the first. The experiments now to be described on the | cooling of wires in liquids will indicate the form of the ev/k term in (16) and also whether the simpler expression is satisfactory. 7 Apparatus. The method of experiment consisted in stretching a wire horizontally at a convenient depth in a vessel full of the liquid under examination, and measuring the electric energy Convective Cooling in Fluids. 927 supply necessary to maintain a measured temperature difference between the wire and the general body of the liquid. The electric energy supplied was determined from the measured current through the wire, and from its resistance as obtained from a Wheatstone bridge. This resistance also gave the temperature of the wire. Kor each of the liquids the cooling of wires of two sizes was studied, the diameters being 0°0083 em. and 0°0155 em. respectively. Both were thought to be pure platinum, but Fig. 1. [011 anya (©) tests of the resistance at the temperature of melting ice and that of steam gave a very low value for the temperature coefficient of the finer wire. The apparatus is shown diagrammatically in fig. 1. The platinum wire TT, immersed in the liquid under examination, formed part of the fourth arm of a Wheatstone bridge, having equal ratioarms PP. By adjusting the slider 8 the 925 Mr. A. H. Davis on Natural bridge was balanced with a small current (0°01 amp.), thus allowing for slight temperature changes of the fluid. An additional resistance dR now introduced into the third arm destroyed the balance, which was then restored by increasing the current through the bridge and thus heating the wire. TT until its resistance had sufficiently increased. Correction for the cooling at the ends of the test wire was automatically effected by having in the other arm of the bridge a shorter piece OC of the same wire* also immersed in the liquid under examination. To be effective, this compensating wire must be greater than a certain minimum length. Ayrton and Kilgour f have given a calculation showing that for a 6-mil wire at about 300° C. temperature excess in still air, the effect of the heavy leads extends about 1 cm. from each end. For finer wires it would be less, and also it would be less if the cooling medium were a liquid, for these carry off more heat than air, so that the end effects become less important. In the present experiments CC was never less than 2 cm., and so the end effect should be entirely eliminated. The energy dissipated in the uncompensated length (/) of TT was calculated from the current (2) through the wire and its (hot) resistance R. The temperature excess (@) of the wire above its surroundings is given by dR/R,a, where Ro is the resistance of the wire at 0° C. and ‘«’ the temperature coefficient of resistance. The heat loss H (in calories) per em. length of wire per °C. temperature excess is given by (ine H —— 4°18] ern 0, e e e ° e (1 7) where 0 =dR/Roa, 4°18 being the factor required to convert watts to calories per second. Details of the wires used are given in Table I. The rectangular vessel containing the liquids under examination had a height of 12 cm. and a base of 17x10 em., and the wires (3 cm. apart) were 3°5 cm. below the surface of the liquid. This vessel required rather larger quantities of some of the liquids than were available.. A smaller one was found with carbon tetrachloride to give the same result as the larger, and so it was generally used * Callendar introduced this method of eliminating end effects in resistance measurements, using it for many similar purposes. Proce. Phys. Soc Lond. xxxiii, p. 187 (1921). + Ayrton and Kilgour, Phil. Trans. A, clxxxiii. p. 3871 (1892). Convective Cooling in Fluids. 929 TABLE I. Details of the Wires used. Rie | Uncompensated og gd’. | length. a. Ro. ; | | a. | rate Ae roan ee at es em. | cis. ohms. “0083 | 5°62 10-* 5:95 ‘00183 2°127 ‘0155 | 3866 x1074 | 6:5 003885 355 | instead. It was 10 em. high witha base 16x6cm. It was used quite full and with the wires 5 em. below the surface of the liquid. In all cases the vessel stood upon an insulated levelling table on a concrete slab on the floor, and no ripples were observable on the liquid surface. It was completely covered in by a draught-proof enclosure of cotton-wool over a cardboard frame, for draughts might by local cooling set up convection currents in the liquid, and might also cause evaporation of the more volatile liquids. A’ thermometer with 1/10° C. divisions checked the temperature of the fluid. Table 1J. is a summary of the physical constants of the liquids used. It was complied from published data in books TasiE IT. + Physical Constants of the Liquids used. Liquid. 104%. | 1042. Ass Uo peo ase Ny cv/k. ac*/k? | Nes 95/ Ny. Toluene ......... 3°42 | 10°99 0:866 0°40 0:0062 2a) wiles | Carbon tetra- chloride ...... 266 | 11:8 | 1°582 07198 | 0-0108 8:03 | 1630 "728 AMINE .......0. 4°] 85 | 1:023 | O°514 | 0:055 69°3 1390 465 live oil......... B02) 70) | O-91o | O47 (1-17 1402 842 363 Glycerine ...... 68 Boi 26M ay) 0 Os 9°3 7940 611 *42* s=specific heat. n =viscosity. o=density. C=sp: v=n/p. * My+10/ My: of physical constants, and since pure liquids were used it is suffiviently exact for present purposes where (as will be seen later) we are concerned mainly with a small fractional + The majority of these values were obtained from ‘ Physical and Chemical Constants, etc.,’ by G. W. C. Kaye and T. H. Laby. 930 Mr. A. H. Davis on Natural power of the values given. Also, as only moderate heating of the wires was involved, little account has been taken of temperature changes of the physical properties of the liquids, though the point is referred to later. The ratio arms were usually 100 ohms eech, 1000 ohms being occasionally used for the larger currents. Hquality of the ratio arms was tested by reversing the connexions and noticing whether the balance was thereby disturbed. To eliminate disturbing resistance changes when the heating current was passed, all connectiny wires were compensated by similar wires in the opposite arms of the bridge, and the resistances R and dR were constructed of manganin as also was the slide wire 8. It was found on test, by removing the platinum wires TT and CC and substituting heavy oil- immersed manganin of the same resistance, that the balance of the bridge obtained with a small current (0 01 amp.) was undisturbed when a heavy current of 1 amp was passed, and that therefore the. compensation was satisfactory. a0c/kv, | H/k. | gda0c/kv.| H/&. Sr 0-0083 0,120 | 1:41 | -0,235 | 1°66 0:0155 0,83 0153 | 1:99 2 0-0083 229 2:55 | 7-90 3°86 0-0155 1-49 3-42 | 51:3 562 Aniline ......... — 0:0083 0225 | 2:04 | 1:21 3:56 | 00155 1:47 261 | 7:9 4-74 Olive oil ...... 0:0083 0,67 | 1:34 | -047 2-18 | 0:0155 0,44 |1:73 | -30 3-07 Glycerine ......; 0°0083 0,87 | 118 | -0,21* | 1-51* | 00155 0,56 | 1:44 | -0134* | 1:99* | { * These refer to 9=2U° C + On the graph of fig. 3 are also plotted the following values for the wires at 2U0° C. excess in air :— Wire diam.... 0°0083 0-0155 gd3abe/ky ...... 5:1210-* 3:00 10-2 EE ie eee, 1:81 2-14 It is seen that the points all le very well on one curve, so that equation (16) appears to represent the results of the present experiments very satisfactorily. ‘The upward exten- sion of the curve beyond the points plotted is the representation on the present basis of the upper part of the curve for gases already referred to. Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3P 938 Mr. A. H. Davis on Natural Fig. 3, - 200 i004 59 4. Gescs QO CCl, 20- x Aniiine Ss e Glycerine r © Olive Oil 6 4. ; ; f | 10000 «575 ‘OD 00001 O00! OOF oO 10 00 1000 10 10 The Cooling Power of Fluids. It appears from the theoretical considerations and experi- mental evidence put forward in this paper and previously, that the convective cooling of similar bodies immersed in fluids may be represented by the formula hL/kO =F (cg L8a6/k*) f (cv/h), and the form of the function /(cv//) is now indicated to be such that we may write the simpler expression AL/ké = FC LP Ogac/ky) . It is to be expected that such a grouping of variables is of more general applicability than for the special case of infinite fluids, especially in so far as it can be obtained from the Principle of Similitude. Where liquids by their natural convective motion carry heat from a hot surface and yield it to a cold one, as with the common case of a hot body immersed in a fluid contained in a cooler vessel, one would in general expect approximate agreement with the formula Convective Cooling in Fluids. 939 if @ represented the temperature difference between the two surfaces. A few experiments would soon test the applic- ability in any doubtful case. For any circumstances to which the formula applies, the form of the function F may be obtained from an experiment in which only one of the variables (say @) is altered. We have some knowledge of its form in certain cases, lor cooling of large bodies in free air or in enclosures, large compared with the size of the cooling body, we may write AL/kO cc (LP @yac/kv)", where x is usually about 1/4, but tends to be as great as 1/3 for large bodies or very hot ones, and to be much smaller for fine wires only slightly heated, as in the present experiments. Measurements of the conductivity of fluids haye been made by methods involving fine wires very slightly heated along the axis of a small metal tube: the validity of this method imphes that n=0 in this extreme case. Since for a given set of circumstances the value of n depends upon the size of the model and upon its temperature excess, it is obvious that the cooling power of a fluid depends upon the circumstances in which it is to be used. Retaining from the above formula only those quantities which relate to the properties of the cooling fluid, we have hc k(ac/kv)”, and for the conditions so far studied n varies from 0 to 1/3, so that the relative cooling power of a fluid may vary between the limits k and (ack?/v)*. Thus it appears for these conditions that the conductivity of the fluid is the preponderating physical property deter- mining cooling power. ‘This result is probably fairly general, for while it may be possible to devise experimental arrange- ments in which & does not enter, suitable conditions for its omission seem generally unlikely to be realized, for it evidently enters vitally whenever the temperature of a particle of fluid (and thus the vigour of its motion and the amount of heat it absorbs or yields) depends upon the thermal conductivity of the adjacent layers. ‘I'he effect of the physical properties, other than thermal conductivity, appears to be such that high specific heat and high coefficient of expansion have the same degree of importance in facilitating cooling that high viscosity has in restricting it. ak 2 940 Mr. A. H. Davis on the Cooling Power I desire to express my thanks to Dr. G. W. C. Kaye and Dr. Ezer Griffiths for the kind and encouraging way in which the facilities for the present work have been provided, to Mr. W. G. Bickley, M.Sc., for critically reading a draft of the theoretical part, and tu my wife for assistance with the numerous calculations involved in the reduction of the experimental observations. May 1922. LXXX. The Cooling Power of a Stream of Viscous Fluid. By A. H. Davis, M.Sc. * [From the National Physical Laboratory. ] iB some previous papers} the author has studied the phenomenon of convective cooling, both natural and forced, from the point of view of similitude, and has shown how excellently experimental data for gases agree with a grouping of variables that Boussinesq { + deduced by hydro- dynamical reasoning for inviscid fluids. The most recent § of this series of papers considered for natural convective cooling the necessary modifications of Boussinesq’s analysis in dealing with the problems of viscous fluids, and the new formula thus obtained was tested experimentally i in certain conditions and found to be satisfactory. The present note develops the theory of jorced convection in the same way, studying the effect of introducing a viscosity term into Boussinesq’s analysis for inviscid fluids. The problem concerns the cooling of a hot body immersed in an infinite fluid stream maintained at a certain tempera- ture, 0 degrees in excess of that of the fluid at infinite distance to which all temperatures are referred. The fluid stream is rectilinear, and moving with uniform velocity »,, at distances from the body sufficiently great. This velocity is supposed to be sufficiently great for the natural convection (gravity) currents set up by the hot body itself to be negligible. We thus neglect the coefficient of expansion of the liquid. Let p and v be respectively the density of the fluid and its kinematical viscosity. Ata given time ¢, and for an element of the fluid at the point w, y, z, let 7, u, v, w, P be the * Communicated by the Author. + Phil. Mag. xl. p, 691% x11, p. 899; xliu. p. 329: if Boussinesq, Comptes Rendus, CXXXll. p. 1882; cxxxili. p. 257 (1901). § See p. 920, of a Stream of Viscous Fluid. 941 temperature excess, the three components of its velocity, and the non-hydrostatic part of its pressure. If /, m,n, are three direction cosines of the general stream of velocity ‘‘v,,” we have as boundary conditions : At infinite distance from) wu, v, w=v,, (l,m, 2), the origin (Ezy): (1) At the surface of thesolid = (w, v, w) =0, id. The hydrodynamical equations of continuity and of motion are Be oF ey 1OP wa yvtu, | O« Oy Oz paw ' es 1aP 19P ; i — = 492, = SE a HV | p Oy p Oz a u’, v’, and w’ being the accelerations of the fluid parallel to the axes. | Let & be the thermal conductivity of the fluid and ¢ its heat capacity per unit volume, then 7’, the rate of change of temper2zture for a given particle with respect to time, is given by POCO NAT ag ens, (3) Also we have U,V, W, T U,V, W, T u',v’, ata 22? iy la eto) Ox OY O(u, U, UW, T) 0(u, U, W, T) +w ye £ >i nc?) The equations in P, u, v, w are everywhere quite separate from those in 6, and hydrodynamieally the problem is the same as that where 6=0, the motion of the fluid being determined entirely by the given general stream and the configuration of the immersed body. Everywhere u, v, and w will be proportional to v,, and P proportional to pv.,,?. Let us endeavour to replace the independent variables t, «, y, z and the functions 7, u, v, w, P by others #,, &, n, ¢, T, U, V, W, IL respectively proportional to them but whose ratios are chosen to eliminate 6, k/c, p, v, v,,. Let us consider the following substitutions : ae eae, 2). (u,v, w)—v_ (CU, V, "o (5) . 9) sor, F=pv,7Ul, t= (v/v,,” ey: 9492 Mr. A. H. Davis on the Cooling Po ioe It is readily found that the substitutions effect the eliminations satisfactorily if cy/k=1. For gases cv/k is approximately equal to unity, and the substitutions should be satisfactory on this ground. Also they are satisfactory if the accelerations wu’ v'w!' of the particle are negligible compared with vV/7u, etc., as would appear justifiable for very viscous fluids. In either of these circumstances the equations take the form (U' V' W’ being omitted unless cv/k=1): 0g ot: aU 8¥ eW_ plot, on ae de tay t ee 7 OF or tee Ol So rg pO | OM oe | ae +(Sa+ 3, 3 e) | Ol an (Om = Com! | on Ge Ses) OL Ow: OW oa (6) | sen tae + on 2 oe | where . | SF rr nari py HO Os Va Wl), 2 OC eee (U,V,WT) =U o - ,woU.¥, WT), oU,V, WT. | Let us put the equation of the solid in the form 3 FU ofP) (2 Y> 2) | =0. : y : ‘ s (7) Thus, if v,,/v changes, this amounts to considering instead of the actual solid, similar bodies having linear dimensions inversely proportional to v,/v.. The direction cosines (,, mj, 7; of the normal will remain the same at corresponding points, and the boundary conditions become : At the surface U=V=W= (0, ie ! At the distance /&+ 77+ C infinite U=l, V=m,W=n. The system of equations (6) and (8) determine U V W II T as functions of &, 7, €, and ¢,, and substituting in the integrals for the new variables their equivalents, we obtain yes TE OEMID SES : ai ay 2 definite functions of ; (9) KOO 2, ¥, 2) 3. 0, OM. of a Stream of Viscous Fluid. 943 The flux of heat h furnished in unit time by unit area of such a body, equal to that which the contiguous fluid layer communicates to the interior of the fluid, is given by Ow 02 Introducing the new variables, we have ih ih ge h= (ke, 6/0) (ao +m +m$p). ae CL) _ At corresponding points of the surfaces /(€,7,€)=0 limiting the bodies considered, the direction cosines l;, m4, n, have the same values, and at corresponding times t,;=const. I M(t, Som, $7 +n, 87). gta EL the derivatives —————~— at corresponding points have also O(E, m £) aT the same values. Consequently, at corresponding times the trinomial coefficient is a function of the shape and orientation of the bodies only. Thus, for a family of similar bodies similarly orientated, having linear dimensions L given by Recev/? eV Alijn=coust., “x. + (12) the heat loss per unit area at corresponding tim es from corresponding points is given by PEMD IO) Wei Ae a) Vat vi 4 ( hOy) and this will also be true if “ h” be the mean heat loss taken at the given instant over the whole surface of the solid. So for bodies of this shape and orientation we have b= (he O/y) fife ott oe (4) which may be written hL wee cw 7e z pF {ese |e, cide AP (ale) When the conditions have become steady—that is, inde- pendent of time ¢,—the formula reduces to AL Oa Ay g =F (=). AUR ae sta (16 And, further, if the conditions do not settle down to complete steadiness, but settle down to periodic fluctuations, then these fluctuations will be similar in form for corresponding cases, and the average value of the heat loss will still be given by (16). 944 Dr. F. H. Newman on a It is desirable to recall two alternative conditions which have been introduced into the proof of the formula. They are that cv/k shall be equal to unity or else very large. If these conditions are not satisfied, it may be shown that the formula becomes hl 7 oes ey ig =! ("s ale re where cv/k is expected to be of little importance in the two extreme cases mentioned. Whether the simpler formula is true for gases on the one hand and for very viscous fluids on the other, and whether — cv/k is important for intermediate circumstances, is a matter for experimental investigation. From data already available it is known that the simpler form is fairly satisfactory for gases, being indistinguishable in this case from the grouping of variables deduced by Boussinesq for inviscid fluids. This is shown graphically in a curve between AL/k@ and v, L/2, given elsewhere *, for the cooling of wires in a stream of air (H/k and vl/vy in the notation of the actual graph). Corresponding data for liquids are not yet available, but an — isolated result has been given by Worthington and Malone Tt for the cooling of a wire in water (cv/k=7 ; v=0:01006), and this yields a result, H/k=21°6, vl/v=38, which is in satisfactory accord with the curve mentioned for air. In the analogous case of natural convection, cv/k has been shown to be of little importance for a wide range of viscous fluids. July 1922. LXXXI. A Sodium-Potassium Vapour Are Lamp. By ¥.H. Newman, D.Se., A.R.C.S., Head of the Physics Depart- ment, Unversity College, Exeter }. [Plate V.] ETALLIC arcs operated in vacua give very intense radiation, and the lines in the resulting spectra are very narrow, whereas with a substance placed between the poles of a carbon arc, working under ordinary conditions, broad lines are obtained, which often show much reversal, the centres of the lines being comparatively faint. This is the case when the sodium D lines are excited, and a bunsen flame, to which salt has been added, is not a suitable source of sodium radiation. As the amount of salt is increased, the * Phil. Mag. xli. p. 899 (1921). Tt Journ. Frank. Inst. clxxxiv. p. 115 (1917). t Communicated by the Author. Sodium-Potassium Vapour Arc Lamp. 945 lines are broadened and show much reversal. Modern prac- tical and research needs require intense radiation and a mono- chromatic source. ‘The chief line must be sufficiently removed from its immediate neighbours, so that if a relatively wide slit is used, other wave-lengths in the immediate vicinity of the one desired are excluded. The quartz-mercury vapour lamp provides such a source; it is easy to construct and work, and does not require continuous pumping to keep it exhausted while running. A sodium vapour are lamp, working on the same principle, has been designed and constructed by Rayleigh*, but it is more difficult to work than the mercury lamp. Iron electrodes are unsuitable, since they fuse and drop off after the lamp has been in use for an hour or two. Tungsten, which seems to withstand the action of sodium vapour, is used instead of iron. This Pigst. lamp requires an applied potential difference of 200 volts when working, although the actual drop of potential across the arc is very much less. The author f has used, previously, a sodium vapour electric discharge-tube which gives intense sodium radiation, but requires continuous heating while the electric discharge is passing. The sodium-potassium vapour arc lamp described in the present work needs no applied heat, can be worked with a small applied potential difference, and requires no attention while it is running. The form of lamp is shown in fig. 1. It is made of quartz, the bulbs A, B being about 3 cm. in diameter, and Joined by a piece of quartz tubing C of internal bore 5 mm. and length * Hon. R. J. Strutt, Proc. Roy. Soc., A. xevi. (1919). + Proc. Phys. Soc. xxxiii. pt. 11. (1921). 0 OQLSQSLS ee eS ee ee 946 Dr. F. H. Newman on a 15 mm. ‘The electrodes are iron rods 4 mm. in diameter, and are sealed with sealing-wax. ‘The current used must be such that these rods never become so hot that the wax is melted or softened. The liquid alloy of sodium and potas- sium—two parts by weight of sodium and one of potassium— is run into the bulb A, and the exit tube D then connected to a glass tap, and the whole exhausted. The lamp can then be disconnected from the pump. An electric discharge is passed through the lamp, the alloy being made the cathode. Under the action of the discharge the oxide on the surface of the alloy disintegrates, and the surface becomes quite clean. Tilting the lamp, some of the alloy flows into the other bulb B. Using this method of introducing the alloy, the part C remains quite free from the alloy. With direct current the lamp works with a minimum applied potential difference of 30 volts, although when once the arc is struck, the fall of potential is only 10 volts with a current of 1°5 amps. ‘The arc is struck either by tilting the lamp in the same way that the mercury lamp is started, or one ter- minal is connected to a small induction coil and a momentary discharge passed. No external heat is required, as that pro- duced by the current is sufficient to vaporize the sodium and potassium. As the temperature rises the current decreases, and the potential difference across the terminals becomes greater. With currents smaller than 2°5 amps. the tempera- ture of the quartz at C is never such that a piece of paper held at this part is charred, and the wax seals do not soften, however long the lamp is working. There is no “browning” of the silica. As the applied potential difference is increased, the radiation becomes brighter, and greater luminosity can be obtained by warming the part C with a small gas flame. This part of the apparatus is, of course, hotter than the other parts, owing to the high current density. The lamp works satisfactorily at any potential between 30 and 200 volts, and the current can be regulated by a resistance in series. If the current rises above 2°5 amps, there is the characteristic ‘browning ”’ of the silica. The lamp does not require con- tinuous pumping while it is working. The sodium-potassium alloy absorbs all gases, particularly nitrogen and hydrogen, while the current is passing. In this manner a very good vacuum is maintained, however long the lamp is in operation. This fact, and the low voltage at which the arc is struck, are two important improvements on the other forms of sodium vapour lamp. The present form will not work satisfactorily with alternating currents. With the lamp it was found that the potassium lines were Sodium-Potassium Vapour Are Lamp. 947 very faint compared with the sodium lines under all condi- tions, and they became weaker as the temperature increased. The relative brightness of the sodium and potassium lines differed in various parts of the lamp, and the subordinate series lines of sodium varied in intensity compared with the D lines, although the latter were always the brightest. At the cathode the potassium lines were very weak, while the subordinate series lines of sodium were strong (PI. V. fig. 1.) At C the radiation was very intense (PI. V. fig. 11.), while at the anode the potassium lines were brighter than they were at other parts of the lamp (PI. V. fig. 11.). The spectrograms were photographed witli a constant deviation type of spectro- meter, Wratten panchromatic plates being used. ‘The sub- ordinate series lines of sodium becaine faint when the current was reduced, and when the lamp was heated externally (Pl. V. fig. 1v.). The electric discharge gave a radiation consisting almost entirely of the D lines (Pl. V. fig. v.). The intensity of the spectrum lines emitted by a mixture of vapours when subject to electrical stimulus depends on the ionization and resonance potentials of the various vapours, and also on the partial vapour pressures. On the Bohr theory the spectrum lines have their origin in the movements of an electron within the atom when it moves from one temporary orbit to another. In the case of sodium the innermost orbit is that represented bythe limit of the principal series —that is, by the term 1°58. The second orbit is represented by the term 2p, and the frequency of the resonance line is that of the first principal line 15 S— 2p. The theoretical value of the resonance potential of sodium vapour is 2°10 volts, and electrons of this energy produce the D lines. Electrons having a velocity corresponding to about 5°13 volts are able to ionize sodium vapour and cause it to emit all the lines, including those of the subordinate series. The resonance potential of potassium vapour is 1°60 volts, while the ionization potential is 4°33 volts. Ina mixture of sodium and potassium vapours, as the accelerating potential is increased, the 7699, 7665 doublet of potassium should appear first, then the 5896, 5890 doublet of sodium, followed by the subordinate series of potassium and sodium respectively. The doublet 7699, 7665 being near the limit of the visible spectrum would be faint, and so it is to be expected that the D lines will, under all conditions, be the brightest lines in the spectra. When the current density is increased, the subordinate series lines increase in luminosity. Sodium vapour having only one resonance potential, the elevation in energy of the 948 Mr. J. J. Manley on the electrons colliding inelastically must be produced by succes- sive impacts or by absorption of radiation of suitable frequency. The electron normally in the 1°5 S orbit of the sodium atom may be forced into the 2» orbit by direct impact. It is possible, however, that the ejection may be brought about by absorption of radiation of frequency 15S—2p. Before it is able to return to the 1°58 orbit and emit this radiation, collision occurs with a second electron, and the electron within the atom is ejected to an orbit still farther removed from the innermost stable orbit It then returns to the 1:58 orbit in two stages, the first step causing the emission of a line in the subordinate series, the second step giving the D lines. The chances of this type of collision occurring will increase as the density ot the free electrons becomes greater. ‘This also explains why the subordinate series lines are very faint compared with the D lines when an electric discharge is sent through the lamp. for the current density in this case is comparatively small. LXXXII. The Protection of Brass Weights. By J.J. Maney, W/.A., Research Fellow, Magdalen College, Oxford”. IXTEEN years ago, I applied to a set of brass weights ; a method introduced by Faraday for protecting iron from rust ; and asthe experiment has been highly successful, other workers may find the plan, or a modification of it described below, helpful. The weights, some of them badly corroded, were first lightly tooled and then suitably polished ; next they were heated in a semi-luminous gas flame until they were nearly red hot, and then suddenly plunged into boiled linseed oil, wherein they were left to cool. The weights having been removed from the oil, were washed with turpentins and then polished with old linen; subsequently they were adjusted, standardized, and set apart for the use of students beginning their course of (Juantitative Chemistry. Now, although the weights have been in regular use throughout the 16 years that have since elapsed, and have been subjected to what may be rightly termed the severest test of laboratory con- ditions and usage, numerous re-standardizations, the most recent of which was carried out a few weeks ago, have con- clusively shown that Faraday’s method for the protection of iron is also remarkably effective when applied to brass, and ~ * Communicated by the Author. Protection of Brass Weights. 949 as a result I find that the original relative values of the whole set of weights are still retained ; no re-adjustment has been required. These observations and conclusions are also borne out by the fact that the uniform and somewhat pleasing bronze-like tint acquired by the weights during the treatment is still to a large extent almost unchanged. To obtain indubitable evidence as to the intrinsic value of the method described above, the plan was lately tried with other weights but without success. And here we may observe that owing largely to the admixture. of one or more sulphur compounds, the quality of the present day coal-gas is very different from that which formerly obtained, and experimental work showed that to this fact must be attributed Fig. 1. my non-success with the second set of weights. The difh- culty which thus so unexpectedly arose, has now been over- come by a method which for convenience and effectiveness leaves but little to be desired. The new plan is as follows :— A ‘“‘vitreosil””? crucible having a capacity of 50 c.c. is loosely charged with asbestos fibre to the depth of about half-an-inch, as shown in the figure. A lid J, also of vitreosil, is inverted and placed upon the fibre; on this lid are arranged triangular-wise three pointed fragments of porcelain or fused quartz for supporting the weight w. The weight having been cleaned and polished and evenly covered with linseed oil, which is applied with the finger and thumb, is stood within the crucible, as indicated in the figure; the 950 The Protection of Brass Weights. crucible is then covered with its lid, and the whole heated with a Bunsen flame. (It is convenient to adjust the flame so that it extends about half-way up the crucible.) During the heating the weight is frequently viewed, and when it has assumed a golden tint the process is complete ; the flame is then removed, and the closed crucible allowed to cool. Finally, the weight is rubbed with an old silk handkerchief and then adjusted. Treated thus, the weight presents the appearance of well-polished and lacquered brass. The protecting film being tough, is not readily defaced, and, if necessary, the polishing may without risk be prolonged : but when the initial operations are correctly performed, nothing beyond a light and brief rubbing is required. In dealing with a number of weights, | great economy both in time and labour may be -effected by substituting for the crucible a vitreosil muffle. The muffle is fitted with a rect- angular tray, which carries the bits of porcelain, placed in groups of 3, for supporting the various weights. The weights having been arranged, the tray with its charge is introduced into the muffle, which is then closed, suitably heated, and subsequently allowed to cool; the weights are then ready for polishing and adjusting. It may be observed that as the supporting fragments are pointed, they but barely engage the surfaces in contact with them ; hence the finished protective film is practically complete. Some experiments were made to determine the average mass of the protecting fil ; the results obtained with two weights, the one of 50 and the other of 20 erms., may be cited. Thecleaned and polished weights were first accurately weighed, then protected and finished as already described ; finaliy they were re-weighed. The larger weight had in- creased by 0020 grm. and the smaller by ‘0006 prm. As the respective areas of the two weights were approxi- mately 17 and 12 cm., we find that the average mass of 1 sq. cm. of film was ‘00012 grm. in the one case and "00005 grm. in the other. It was found that the smaller value most nearly represented the weight of a normal film ; the other and larger value was exceptional. Daubeny Laboratory, Magdalen College, Oxford. Peron ho Al LXXXIII. Note on the Analysis of Damped Vibrations. By H..8. Rowe *. HE two primary difficulties in the analysis of damped vibrations are the nature of the friction and the position of the zero. In most cases it is sufficient to assume that the friction is a combination of so-called solid friction—a constant, and of fluid friction, proportional to the velocity. In the ordinary view it is inconceivable that these two kinds of friction can coexist at the same time and interface, for the conditions supposed to produce these two kinds of friction are essentially different ; dry and wet, or molecular film and measurable film. In practical cases, however, the two kinds of friction can coexist in a system as, for example, where a body slides or turns on dry surfaces and is damped by fluid friction. Thus the equation of motion may be taken as make ee PaO fg ws (LL) where the signs of F and of « are the same ; put 2=X>+ EF /e’*, and the solution of equation (I.) is gig Ane 2" cosin'ts 2 (AL) where : n= Vn? —k?/4m? and ae write ies and ka/2mn'=r; so that e*=6, where 6 is the logarithmic decrement for half periods. Assume that a datum line is drawn at a distance E trom the true time axis, and let R; be the reading from this datum corresponding to the ith half swing ; then a ee. (LY By — HS Age, ee C2) ip AN So ee we (BD ieee gS). SY dg AAD Ory: eth is) @ Seen * Communicated by the Author. 952 On the Analysis of Damped Vibrations. Subtracting (1) from (2), and Risky = Ay +0)... Si Subtracting (4) from (2), and RyRy = Apo(l = 0"). eee Subtracting (3) from (5), and : Ry— Ry = Ayd?(1—8"). se SD Dividing (8) by (7), and Ree R, SO... Cae ee nn Substituting in (6), we have the first amplitude : Ay = (R,—Ry)/1+6 _ (Ri—Ro)(Ri— Ry) a Dividing (7) by (6) and adding (2) and (3), we have ie Lo Bie oR (Bei ea which gives the position of true zero. The solid friction term § follows from (1), and since ¢? is easily found statically, F the solid friction of the system per unit mass follows by division. Where the system is dead-beat, the foregoing method does not apply, and one way of solution is then by tuning of the system by adding mass or increasing c? or both so that sufficient equations are determined tor elimination. The curve, of which the vibration in II. is a projection, is an equiangular spiral with alternating origins distant 2K /c? apart, and it may be traced in either of two ways according to circumstances. In the first place an arithmetic spiral (see Phil. Mag., July 1922, p. 284) may be drawn and the radii vectores shortened logarithmically, or an equi- angular spiral may be drawn and portions taken out each sub- tending a, and such that the initial radius vector of one por tion is 2F/c? less than the final radius vector of the preceding portion. Clearly the parts run smoothly together on account of the equiangular property of the spiral. Here it may be added that since the evolute of the arith-. metic spiral is a straight line 2F/c? long, the curve can be drawn mechanically by coiling a fine thread round two pins 2F'/c? apart. ‘The same curve is described by the hand of the housewife in winding up a card of “ mending.” Liffect of Variable Head in Viscosity Determinations. 953 The outstanding difference between solid friction and fluid friction in damped vibrations is that in the one the dissipation per cycle is proportional to the amplitude, whereas in fluid friction the dissipation varies as the square of the amplitnde. Since the energy of the motion varies as the amplitude squared; the dissipation per unit time with fluid friction is a constant fraction of the energy of the system, whereas with solid friction the dissipation as a fraction of total energy is inversely as the amplitude. Hence in solid friction the rapid damping of small vibrations until finally the dead region of width 2F'/c? is reached. Clearly in the two systems the envelopes of the two vibration curves may be tangential, in which event solid friction may be mistaken for fluid friction. As the foregoing analysis indicates, five half vibrations or two complete periods suffice not only to safe- euard against this possibility, but also to apportion the relative magnitudes of the two sets of frictional forces. It seems possible that some such analysis as is here outlined may be a useful instrument of investigation in connexion with friction and lubrication, affording at least some sort of criterion in so-called border-line cases. LXXXIV. The Full Effect of the Variable Head in Viscosity Determinations. By Frank M. Lipstone*, late the publication in this Magazine of my paper on h the Measurement of Absolute Viscosity (February 1922), it has been pointed out to me by Mr. W. H. Herschel, of the American Bureau of Standards, and by Dr. Guy Barr and Mr. L. F. G. Simmonds, of the National Physical Laboratory, that the logarithmic head correction in the ‘“‘ viscous ” term of the equation is, strictly speaking, incomplete, inasmuch as it is based on the assumption that the head varies directly with the velocity. Barr also makes a necessary correction in the final kinetic energy term of the approximate equation, which should read as in equation (2) below. As no attempt appears to have been made to finda general equation embracing all these corrections, it is here proposed to try to find the exact expression, however laborious and cumbersome, in order to ascertain to what extent the results obtained by means of the ordinary formule deviate from the true value. The premises of the whole argument are in- cluded in the generally accepted equation, __aregpth Vp SEO GEND Salt” * Communicated by the Author. Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3Q 954 Mr. F. M. Lidstone on the Full Effect of the the coefficient of the K.E. term being taken as unity. This equation is true provided the head h is constant ; but unless some compensating mechanical contrivance is used, such as that adopted by Hyde (Proce. Roy. Soc. A, xevil. 1920), this condition is never absolutely realized in practice, since no matter how the pressure is applied or maintained, as soon as flow starts, there is a change in the hydrostatic head and the total head becomes at once a variable. We have then, by making dt depend upon dh, first to integrate the whole expression with respect to A over the interval H—F. : A Putting SEE ie Man) Ris Te Shas Sarl eA Le ie Ah(H—F )dt nd Bdh . dh ~ (H—-F)dt’ . I~ PD 3° ae (8 areata VEBy/ 1+ yop SAMHOR) = AC The plus sign being obviously the only one permissible, we ; . oe ' pel | get, writing C for LAB? Hd /B HV/et dt= onl — $$ “de, { 2A(H — In 2 / A(H—F) ps Ey which, after a little manipulation, gives finally 7 = TI'Jp Get — F) = \ i ar H =e 8Vi{log,.v ta vo) ty / 1+ G-a/ 4G} (1) Now, since C contains 7, to evaluate 7 from this expression _would lead to a number of very unwieldly power series. However, it will be seen that C must be large in comparison with H or F'; hence a small change in C will not greatly. affect the result. As a first approximation, then, we can take C as equal to mw rg?(H— EF)? 4V*(log. H/T)?" Calling this Cj, we get a value for 7 which we will call 7. ny 16? CG ‘ gp? Or Uo. We can now get a closer value for CU, namely Variable Head in Viscosity Determinations. 955 Re-calculating with this value we get m2, which gives us from which 7; is calculated, and so on. The first value for C or C, being too large, 7; will be too small. Simi- larly, 7. will be too large and 73 again too small. Thus 7123 .... form a series in which the even terms and the odd terms respectively converge asymptotically to the true value for 7. The rapid convergence of the series con- siderably shortens what would otherwise be a_ rather laborious calculation ; three or four terms are sufficient to fix the final value. As an example, we take one of the standard runnings of water at 20°C., and select in particular that one as set forth in detail in Archbutt and Deeley’s ‘ Lubrication and Lubricants, 3rd edition, p. 157. This example is chosen as there is a considerable fall in the head, although, on the other hand, the kinetic energy term is small. We quote the data of the experiment in full :— H= 23:56 | g = 980°51 F= 11:60 | p= 099826 r= 00309 V= 4:00756 t = 136-0 I= 21-991! From these figures we get as the first value for C or C, = 890°52 giving 7, = °0100609, feom which C,= 897°24 ..,° 7, = °0100692 s Cz = 880°69 F455 gy = 70100680 5 ©, = 88049" % 5, oy, = 0100682, from which we can write down the final value of 7 as -01006818. The value of 7 calculated from the same data, using the equation arrget( Be Vie CH? — *) log, H/F 9) "= 3Vilog, H/F Wnu(—F)(H+F)’ °° & is 0100721. The difference between this and the correctly calculated value ‘01006818 is not very striking, but when one con- siders that here the kinetic energy term is small (only ‘5 per cent. of the total) this is not surprising. In order adequately to demonstrate the shortcomings of the ordinary formule, it would be necessary to make a running under a considerably increased initial head, in order that the K.E. correction should become appreciable, and to continue the running until the final head was relatively small. 3 Q 2 | 956 | LXXXV. Quantum Theory of Photographic Exposure. (Second Paper.) By Vi. SUGBERSTHIN and Ay) ieebe TRIVELLI™. N the present paper an account will be given of some additional experimental tests of the light-quantum theory of photographic exposure proposed in the first paper of the same title t, and some further theoretical formule will be deduced from the fundamental one given in that paper. Virst of all, however, due mention must be made of certain very valuable experimental investigations, since published by Svedberg, which seem again to corroborate the theory, also of a paper by Svedberg and Andersson published somewhat. earlier, but not brought to our notice until the first paper had been dispatched for publication. 1. Concerning “The Effect of Light,’ Svedberg and Andersson’s paper (Phot. Journal, August 1921, p. 325), dealing under that head with only a very few size-classes of grains (each class, moreover, of a very considerable breadth), contains only the qualitative though definite conclusion that ‘for equal exposure the percentage of developable grains is always greater in the class of larger grains.” The quanti- tative, viz. exponential dependence of this percentage upon the size (area) of the grains, is obtained and well verified experimentally in the case of bombardment by erays, Kinoshita’s experiments of 1910 having made it very probable that each silver halide grain hit by an a-particle is made developable. The latter being granted and the discrete nature of a-rays being a palpably established fact, the validity of the exponential formula, in our symbols k=N (1—e-”*), had to follow as a necessary consequence. Its verification is properly a verification of Kinoshita’s statement, and by having thus tested it experimentally Svedberg and Andersson have certainly done an important piece of work, especially as Kinoshita’s result was contested by St. Meyer and v. Schweidler. In the next section | analogous experiments with @-rays are described, but the results thus far obtained are not conclusive apart from enabling the authors to state that one or two -particles striking a grain do not as a rule make it developable. Finally, returning once more to the eftect of light (p. 332), * Communication No. 149 from the Research Laboratory of the. Eastman Kodak Company. Communicated by the Authors, + L. Silberstein, Phil. Mag, July 1922, p. 257. Quantum Theory of Photographic Exposure. 957 the authors remark only briefly that an analogous concep- tion might also assist in the interpretation of the mode of its action ; but add that if the quantum hypothesis be assumed, ‘“‘ the difficulty arises that the real blackening curve has not the exponential form prescribed by this hypothesis if we suppose each halide grain to be made developable when struck by a single light quantum.” ‘They seem to forget that the simple exponential formula yielded by a quantum © theory relates to the case of equal grains, which is not that of real emulsions, and that in order to obtain the blackening curve (say density J) plotted against the logarithm of exposure) that elementary formula has to be integrated over the range of sizes, which presupposes the knowledge of the frequency curve of the emulsion, and the somewhat intricate question of the relation between the photographic “ density ” and the total of blackened aréas has to the treated*. ‘he latter question, simple though it be for one-layered coatings, becomes particularly mnie in the usual case of many layers of grains. It is for this reason that the best way of testing a similar theory consists in microphotogr aphic counts and planimetric measurements of the individual grains. At any rate, Svedberg and Andersson propose to turn to another more complicated assumption f which, they expect, “ will actually predict a blackening curve of S- shape.” They propose to discuss this possibility on another occasion. The second of the papers alluded to, due to Professor Svedberg himself (Phot. Journal, April 1922 , p- 186), has a more direct bearing upon our subject, and may turn out to supplement our-own tests by offering, as it were, an inter- mediate link in the conjectured mechanism of the action of impinging quanta or light darts. In this paper Svedberg proposes to explain the behaviour of the grains noted in his preceding paper by a single hypothesis, and to test the latter directly. His hypothesis is that the product of the light action on the halide grain consists of ‘‘ small centres distri- buted through the grain or through the light-affected part of the grain according to the laws of chance,” and that a grain will become developed if it contains one or more such centres. If v be the average number of centres per grain, the per- centage probability that a grain will contain at least one centre (and will therefore be developable) is P=100 (1—e7). * Concrete examples of such a kind will be treated in the third paper on our subject. + Namely, that a certain minimum number of quanta must strike the grain within a certain maximum part of its area in or der to “build up a silver nucleus large enough to act as a reduction centre.” 958 Dr. L. Silberstein and Mr. Trivelli on the Now, it would be enough to assume that these centres are produced by discrete light-quanta impinging upon the grain, and the formula P=100 (1—e~™) would follow at once. (For, if n be the number of light-quanta per unit area, and a the area of a grain,v=na.) But Svedberg does not make this assumption *, and devotes instead the remainder of his paper to testing directly the above formula for the occurrence of at least one centre and the corresponding chance formula for the percentage number of grains having no centres, of those having one or two or three centres, etc., having succeeded in making these centres or, in Svedberg’s own words, ‘‘ the nuclei corresponding to the developable centres,” visible and accessible to measurement. For details of these elegant experiments the reader must be referred to the original paper. Here it will be enough to say that the recorded “dots” or visible traces of those centres were found distr?- buted very much in accordance with the probability formule, namely, in one experiment with light and one with X-rays. Only two size-classes of grains were treated in each of these experiments, and with regard to the dependence upon exposure Professor Svedberg (p. 192) has thus far only roughly stated that the percentage number of developable grains “increases approximately exponentially as function of exposure,” at least for normal and for over-exposures in the case of light (and probably for all exposures in the case of X-rays) though not for under-exposure to light. The paper is concluded by the remark that to account for the deviation from the exponential formula in the case of low light-exposure, we should probably haye to adopt the quantum point of view, and that in the case of light (a quantum of visible light containing 5000 times less energy than an X-ray quantum) “ several quanta would have to be absorbed very near one another to forma developable centre ”” Such a view, however, can easily be shown to be untenable. At any rate, Professor Svedberg proposes to test it by experi- mental investigations which are planned in this direction. * In the discussion which followed upon the reading of Svedberg’s paper, Prof. T. M. Lowry mentioned such an assumption of a ‘‘ bombard- ment by light corpuscles” as the simplest interpretation of Svedberg’s photographs (of the “‘ centres”). Other speakers, however, were rather hostile to such a view, and Mr. B. V. Storr considered it even equally conceivable that the “ centres” distributed haphazardly might be present before the light action, but such a state of things would have hardly escaped Svedberg’s notice. At any rate, Professor Svedberg will no doubt meet Mr. Storr’s objection by appropriate control experiments. Control experiments of such a kind, viz. desensitizing experiments, are now being made by Sheppard and Wightman. Quantum Theory of Photographic Exposure. 959 The existence of the aforesaid “centres”? as seats of incipient development, around which the developer's action gradually spreads, has been known for some years *, and has been observed, among others, by Trivelli. But the important discovery that these centres are haphazardly distributed is entirely due to Professor Svedberg. If his results are ultimately confirmed by further experiments, especially for a series of different exposures, it will be possible to consider these centres as an intermediate link in the theory proposed in our first paper (the centres marking the spots where the grains were hit by the light-darts). Im the meantime, however, our further tests have to be conducted by con- sidering the lasé link of the chain, 7. ¢., by counting the grains of each size-class affected and ultimately developed. 2. Before passing on to the description of our further experimental results, a few words must be said in defence of the property attributed in our first paper to clumps (aggre- gates) of grains which apart from some single grains con- stituted our chief material, An explanation seems the more necessary, as another recent paper by Svedberg + contains results apparently clashing with what we believe to be the behaviour of clumps with respect to light. The property assumed by us, as the expression of experimental facts, was that a clump, 2. e., an aggregate of grains in contact with one another, behaves as a photographic unit, by which is meant that if any one of its component grains is made developable, the whole clump will be reduced by a sufficiently long develop- ment. We have since been able to test this behaviour in a variety of ways. On the other hand, Svedberg concludes from his experi- ments that there is no transference of reducibility (develop- ability) from one grain to another ‘‘in direct contact’ with it. (See especially p. 184, loc. cit.) This apparent discrepancy seem to be due to the circum- stance that Professor Svedberg worked with an emulsion (a single kind only) consisting of rather small and almost spherical ¢ grains, whereas our “material, and especially the so- called W-12-C experimental emulsion, with which all the work in question is being done, consists predominantly of large and very thin, flat polygonal plates or tablets which are in mutual contact either along a whole edge or, still more intimately, are partly piled upon or overlapping each other. * Of. M. B. Hodgson, Journ. Franklin Inst., November 1917. + On “The Reducibility of the Individual Halide Grains,” Phot, Journal, 1922, pp. 183-186. 960 Dr. L. Silberstein and Mr. Trivelli on the The fine spherical grains of Svedberg could have only at the utmost a point contact, and this might not have been intimate enough. It is even credible that in view of the Brownian motion of these minute bodies there was actually no permanent contact between them, as becomes very likely from Svedberg’s remark on page 185, that ‘“‘even over such a small distance as 1 micron no noticeable transport of silver ions takes place.” At any rate, we have found in our case the property of clumps as units well verified. Without attempting to reproduce in this place all our available evidence *, we may support and illustrate the said principle by the following data. Fig. 1 represents the frequency curve and, in the Janey, alo 00 800 wi2C a 3154 GRAINS MEASURED. 5.71 - 1(0°GRAIRS PER SQ.CH. PLATE=H. X= 1.04 p7CALC. X=i.01p708s. 175 700 o=SiZE—FREGUENCY PER 1000 GRAINS, OBS. A=S!ZE—FREQUENCY PER 1000 GRAINS,CALC. Y=120e@-081%-0.8d? uiis0 600 B=SIZE-AREA PER S0.cM. PLATE. Y= -0.8Kx-082 = ——— s ome C= T0060 cM. P id un GRAINS. uo re) i2) PER SQ i000 = ow w& 8 A-FREQUENCY PER ‘ 7 wn KS) 3 < @ B-AREA IN {2 x 10-8 Ww wu << 9 inset, a microgram of a sample of grains of the aforesaid W-12-C emulsion. This emulsion was spread over the glass plate in a single layer so as to obtain the maximum number of grains per unit area with the least possible overlapping. The emulsion, after the coating, was kept in its liquid state long enough to enable the majority of the grains to settle with their flat faces on the surface of the glass. Under these circumstances they, and especially the larger grains, form numerous clumps of from 2 up to 33 grains, as will be * Discussed in a paper just sent to Phot. Journ. by Trivelli, Righter, and Sheppard. [This paper has since been published in Phot. Journal for September 1922, p. 407. |] Quantum Theory of Photographic Exposure. 961 manifest from Fig. 2, curve marked VV. After exposure and development the clumping of the survived grains was determined all over again and is represented by the curve marked N—K; the curve marked K is the difference of these two curves and represents the clumps affected by light. Fig. 2. Wi2C CLUMPING CURVES N=ORIGINAL CLUMPS PER SQ.CM. ONELAYER PLATE. K= AFFECTED CLUMPS PER SQ.CM. ORELAYER PLATE. K,=Ny ((— @-0.9938) N—-K=REMAINING CLUMPS PER SQ.CM. La K.= Wreacs°sn = 8000 Km =~ @-0.4538 Na @ MaNUMBER OF GRAINS IRA CLUMP, w °o ° ° CLUMP-—FREQUENCY PER SQ.CM X 107? ON ONE LAYER PLATE N 8 °o This would suffice perhaps by itself to show that our clumps behave as photographic units. But additional evidence is afforded by figs. 3a and 30, in which all the individuals were carefully blackened by hand on a microgram originally enlarged 10,000 times ; the former of these figures refers to the original unexposed one-layer grains, and the latter to 962 De. dh. Shenae and Mr. Trivelli on the the grains surviving after exposure and development. A glance will show that the majority of clumps, and especially the larger ones, are removed entirely. Of such pairs of samples as figs. 3a and 36, about forty were made, and the behaviour was always typically the same.’ Further and more direct experimental tests of the adopted clump principle are now in progress, notwithstanding that we have but little doubt about its correctness, always, of course, in relation to the material which we are using. And we feel sure that the same principle can be firmly relied upon in what follows. 3. Let us recall that the theoretical values of the per- centage number y=100k/N of clumps affected, as given in the fourth column of the table in our first paper, were calculated by means of formula (12), log (1-4) = —nal1— /clals er with the values of the parameters = ie BY 2 eer . ae a= 0091 Ww: the meaning of all the symbols being as before. The agree- ment of these values with the observed ones, ranging over 33 classes of grains and clumps, was excellent, thus proving, at any rate, the essential correctness of the formula as far as the dependence on size (area) a goes. To test it with regard to the exposure or n, portions of the same plate were subjected to the action of the same lght source, ceteris paribus, for one-half, and for one-quarter of. the time of the original exposure. The same method of evaluating WV and k being adopted as before, the results tabulated below under yop5. were obtained. Now, without even taking the trouble of retouching the values of the parameters in adaptation to the new observations, o was taken as in (12a) and n equal to one-half and to one-quarter of its original value, respectively. Since the exposure is, at any rate, proportional to n, our formula with these n-values should represent the two new sets of observations. The following table gives in the first row the number of grains in a clump *, and in the second row the average area a of each class of clumps, in square microns, as before; the third and * Starting from 2, since with these weaker exposures reliable counts of single grains affected could not be secured. . Quantum Theory of Photographic Exposure. 963 the fifth rows contain the percentage numbers of clumps affected calculated by (12) with c=0°097 and "572 Gr) Gules ~ =0°286 per pu? : 2 and (IT.) we Ae =('143, respectively, and the fourth and the last rows those observed. The last but one column refers to clumps of 12 and 32 grains, and the fact that almost all of these have been affected (Yovs. =100) gives an additional score of verifications of the theory (though in the case of (II.) the observed “100” sets in somewhat too soon). Grains in Clump 2 3 + 5 6 7 8 9 10 1 itera ae 173 303 488 62 74 86 98 Il 12 13 ...>25 (Veale, 282 442 643 743 81-0 859 89-7 924 942 955... 998 ‘Wenn, 21 57 638 745 875 96 97 97 965 100 ... 100 Beas foe, 40 749s) bb 626 Gre 2735759 TES... 95-7 a 13 876 423 53 66 825 865 (?) 894 100 ... 100 The agreement, although in general not so close as in the previous case, is certainly satisfactory and in three or four instances even remarkably good. Notice especially the case of four-grain clumps which show perfect agreement in all three exposures, the calculated and observed values in the original exposure (cf. first paper) having been 87:3 as against 87:1, and now 64°3 and 41:2 as against 63°8 and 42°3. Almost the same is true of the five-grained clumps. But in- general the agreement is good enough throughout the array of clumps *. 4. Notwithstanding the good agreement and the consistency of these three sets of results with regard to the values of n and o, some critical remarks must now be made about the meaning of the latter parameter. It will be remembered from the first paper that o or mp? was originally introduced as the (average) “cross-section ” of the light darts, and p as their equivalent semi-diameter, and the mathematical réle of this finite diameter was fixed by assuming that a grain is made developable only when it is “fully ” struck by a light * The outstanding discrepancies being attributed mainiy to the uncertainty of the (average) sizes a of the clumps and perhaps also to disregarding the effect of tue finite range of a within each class of clumps, How this finite breadth of the classes can be taken into account will be shown presently. 464 Dr. L. Silberstein and Mr. Trivelli on the dart. This gave as the efficient area of a grain, instead of a=Tr"*, a =al1—p/r|?. Now, exactly the same formula would arise if we assumed that, no matter what the thickness of the light darts (and. whether it is finite at all), a grain is made developable only when the aavs of the dart hits it in a point not too near the edge of the target (grain), thus excluding from the total area a boundary zone of a certain breadth p. Such a condition is not altogether fantastic, and one might support it by imagining that if the grain is hit too near its edge, an electron is still ejected and a “centre” of reducibility is produced at the spot, but the wave of development, stopping dead at the edge, has not such a good chance to spread over the whole grain as when the centre is well within the target. If so, then the empirical principle that a grain is either not affected at all or is made developable entirely would require a qualification, viz., the exclusion of that boundary zone. This alternative, therefore, should and can still be tested. If it is supported by experiment, the original interpretation given to p or o can be abandoned, since it certainly is not very satisfactory. Not that there is anything incredible in the light darts having a finite thickness and a cross section such as one-tenth pu”; so far as we know, they may be trains of waves of even much larger transversal dimensions. But the unsatisfactory point about this interpretation is that it is hard to imagine why the grain to be affected at all, 2. e. to have a photo-electron ejected, has to be hit by the whole of that cross section. For, if so, then, unless some light darts have a diameter of the order of 107° cm., no such things as simple atoms or molecules could ever have their electrons ejected by light*. Yet, a grain, as a crystal lattice, may, after all, behave as a single molecule, at least in the present connexion, and the original réle attributed to the cross- section of the light darts, though repugnant, may still turn out to be a useful working hypothesis. To ensure the possibility of being fully hit and therefore affected, even to the smallest available silver halide grains, it would be enough to treat o in our formula as the average taken over a sufficiently ample interval of sections down to very small ones. It would be premature to enter into quantitative details of the consequences of such an assumption. But it seems proper to mention even at this stage that an assumption * Whereas the photo-electric effect has been obtained with gaseous substances, though not beyond every doubt. Quantum Theory of Photographic Exposure. 965 of this kind can well be tested experimentally. In fact, if that assumption be correct, then the light traversing two or more equal photographic plates piled upon each “other should contain, successively, a larger percentage of the coarser light darts, so that the formule of type (12) repre- senting the number of affected grains or clumps of various sizes should have not only a decreasing n, but also a succes- sively increasing average value o of the cross-sections of the darts, a comparatively larger proportion of the more slender darts being absorbed each time. In short, we should have a kind of sifting effect. Such experiments which, to be at all convincing, require obviously a much higher degree of accuracy in counts and area measurements, are now in preparation. Their results will be published in a subsequent aper. In the meantime, the parameter o may and profit- ably will be retained as a small but desirable correction of the exponential formula without, however, being given either of the alternative interpretations. It may be well to add here also a few remarks about n, the chief parameter in the fundamental formula. This was originally defined as the number of light-quanta or darts thrown upon the photographic plate per unit of its area. Now, apart from the generally small correction term con- taining o, the parameter n appears in the formula only through the product p=na, where a is the area of the grain. Thus, essentially only the value of this product (a pure number) can be determined from microphotographic experiments. Suppose now that the sizes of all grains of the given emulsion were reduced in the same ratio, converting every a into ea; then, the same experimental value of p would indicate a number of light darts : times larger. Now, such would exactly be the position if for every grain not the whole but only a fraction e of the area were vulnerable, 2. e. deprived of an electron on being hit by a light dart. The grain may be sensitive only in spots scattered over its area, “and each perhaps of very Gonats dimensions.. Provided that all theso spots occupy a jived fraction e of the total area of the grain, the micro- hotographic counts and measurements could not inform us about the value of this fraction unless the exposure given to the plate is known in absolute energy measure. ‘Thus, for instance, if, as was tacitly assumed, e=1, the number of light daris in the set (1.) of observations just described would lead 966 Dr Le siterctenaad Mr. Tavelloae to n=0'286 per pw? or about 29 million darts per square centimetre of the plate; but if, say, only one-thousandth of the area of each grain were vulnerable, we should conclude that - 29 milliards of darts were thrown upon each cm.’ of the plate. But it would be idle to speculate upon this subject and, as far as we can see, the only way of deciding whether that suggestion Is correct or not and of deter mining the value of | the fraction e¢ is to measure the exposure energy in absolute units *. Now, in none of our experiments thus far reported ~ was the energy value of the exposure even roughly estimated, not to say measured. But in order to decide this important question, preparations for measurements of this kind are now in progress in this laboratory, and their results will be published in due time. 5. Liffect of finite breadth of size-classes of targets.—The short name “target ” will now be used for either a single grain or a clump of grains in suthcient contact to act asa photographic unit. In the three sets of observations hitherto reported, the targets were classified according to the number of grains contained in them (from 1 to a3 and for each class the average size (area) was used as a in the theoretical formula, without taking account of the finite breadth of any such class, 7. e. of the interval, a, to a, say, over which its indi- viduals ranged. It was ‘not possible with the said classi- fication to secure reliable estimates of this breadth, which, however, for some classes might have been considensnle (perhaps of the order 1y?), and ‘at any rate varied from class to class. It is likely that some of the outstanding dis- crepancies are due to these neglected tactors and especially to the latter. To eliminate this source of error, and at the same time to avoid the laborious planimetrization of targets within very narrow limits, we propose henceforth to divide the whole material of targets into deliveruiely broad classes, all of equal breadth, say 2a. ie then, the average size of any of these classes of targets is Tee as he variable ain our formula, a correction has to be made for the finite value of 2a. This correction can easily be found. Disregarding for the moment the o-term, the number of targets of a class of breadth 2«=a,—a, affected by n darts, is * Although even then the final result would be made doubtful by the uncertainty “whether the total light energy (as required by Einstein) or only a fraction of it is conveyed in discrete quantum parcels. Quantum Theory of Photographic Exposure. 967 by the fundamental formula (7), first paper, k=\ (a) [1—e-™|da, where f(a)da is the number of targets of size a to a+da originally present. Now, if 2 is of the order of 1/2 or even ly’, we can take f(a) =const. within the integration interval with sufficient accuracy for all of our experimental emulsions. Thus, denoting by V the original number of targets in the whole class, so that ie) OND k ena e Raz ve shall h — =l|— we shall have N DAs : or, writing simply a for the average @=1(a,+a,), and there- fore, dg=a+a, a,=a—4, . i NO. op na k —|—e-™. ae a ae N 2ne Remembering that 3(e"*—e~"*)= sinh (ne), writing for brevity N v=log Pen and replacing a in the chief term by ae we have ultimately the required formula sinh (ne) ; Na (13) v=na'— log (14) Notice that the correction term depends only on nz, that is to say, for X=const., on the product of the exposure and the class breadth. If this product is a fraction, such as one-half — or even two-thirds f, we can write, up to (n«)‘, em es (CLAY If, as explained, all the contemplated targets of the emulsion are divided into classes of equal lreadth 2a, the * This is accurate enough provided o/a is small. In the correction term the semi-breadth a requires practically no amendment. + If a=1p” and the exposure is as in the previous concrete cases, the value of this product does not exceed 0°6. 968 Quantum Theory of Photographic Exposure. correction term in (14) is, for a given exposure, constant throughout the array of classes, and v plotted against a should give a straight line. If o were non-existent or negligible, we should have a straight. line for v plotted against a, itself. The aforesaid classification of targets and the corresponding formula (14) will be used for analysing all the experiments now in progress. For the present, we are able to quote only one such set of results condensed in the following table. The targets (grains and clumps alike) were all divided into five classes of equal breadth 2a=0°60y?, ranging from 0°20 to 0°80, from 0°80 to 1°40, etc., as shown in the first column, which gives the average sizes @ in square microns. ‘The third column gives the observed number of targets surviving for every NV targets originally present, each of these data being an average of counts on four different domains of the plate. The fourth column contains the percentage number y=100 5 calculated by (14), to wit, with of grains affected, as observed, and the fifth, as n— 0°25) per 27, o=0°0081p7. a. N. N—k. y obs. ycale. A. 0:50 190°3 173°3 8:9 9°0 —O0:1 1-10 140-0 103°5 26°0 (20°6) (+5°4) 1-70 62°6 43°0 313 311 +02 2°30 314 18°7 40°4 40°3 +01 2°90 19°8 10°3 480 48°4 —0-4 The agreement is, apart from the second class, bracketed as an “outlaw,” almost perfect. The ‘‘ cross-section” of the darts, or what o may stand for, is agreeably about ten times smaller, 7. e. the diameter three times smaller than that previously obtained with the same light source. ‘This is not to say that the reality and réle of o is herewith settled. Yet it is interesting that without o, that is to say, with a’ in (14) replaced by a itself, no choice of n yields such a close agree- ment. As to the correction term due to the finite class breadth, it may be mentioned that in the present case it amounts (as a subtrahendum from v) only 10 0°030. Rochester, N.Y., June 23, 1922. [ 969 J LAXXVI. The Discharge of Air through Small Orijices, and the Entrainment of Air by the Issuing Jet. By J.8.G. Tuomas, D.Sc. (London and Wales), A.R.C.Sc., A.LC., Senior Physicist, South Metropolitan Gas Company, London*. [Plate VI.] INTRODUCTION. 4 Naas present paper details some of the results obtained in a preliminary investigation of the conditions deter- mining the entrainment of air by jets of various gases. As there exists considerable uncertainty as to the representation by means of a formula of results for the discharge of gases through fine orifices T, it has been considered desirable to include also a short discussion of results, under this head, obtained during the work. In the case of the entrainment of air by a jet of gas either lighter or heavier than air, the volume of air entrained per unit volume of gas in the jet is ditferent according as the jet is directed in an upward or downward direction. With a jet of air, such an effect, if present at all, is small. As the great majority of the practical applications of air-entrainment by gas jets issuing from small orifices with which the author is concerned refer to downwardly directed jets, it was decided to confine the preliminary experiments with air to such direction. HXPERIMENTAL. (a) The Discharge Tube-—The flow systems employed in the present investigations are shown in fig. 1, Air was delivered under constant pressure to the brass tube A, the interior surface of which was carefully smoothed, and issued as a jet from the orifice in the disk B, situate at the lower end of the tube. Particulars of the preparation and mount- ing of the disks and orifices are given later. The tube A was provided with a water jacket through which tempered water flowed, so that throughout the whole series of experiments the temperature of the water in the jacket was maintained constant at 13°C., with a possible variation of ‘5° C., the temperature being taken by thermometers placed at the inlet * Communicated by the Author. + See, e.g., Buckingham & Edwards, Sci. Papers, Bureau of Standards, vol. xv. pp. 574 et seg. (1919-20); Walker, Phil. Mag. vol, xliii. p. 589 (1922), Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3R 970 Dr. Thomas on Discharge of Air through Small and outlet to the jacket. The pressure of the air in the tube A at a distance of about 1°5 inches above the orifice was determined by means of the tube C, which was closed at its lower end and provided with a number of small circular openings on its cylindrical surface near its lower end. The tube C was connected with a water manometer, which was read to 1/50 mm. by means of a cathetometer microscope. Calculation showed that the deficit of pressure due to motion of the air in A was ‘negligible. Air delivered from the orifice B passed downwards through the tube D, of vertical length about 10 inches, and bent as shown. ‘The lower end of D was shielded by a large open box, so that disturbances RG in the air of the laboratory were very largely prevented from affecting the jet*. No Venturi tube, to increase the air- entrainment, was inserted in Din the present series of experi- ments, as it was found that the same tube was not equally suitable for use with all the orifices emploved. (b) The Air Induction Tube.-—-Air set in motion by the action of the jet issuing from the orifice flowed towards the jet through the flow tube EH, in which was placed a hot-wire anemometer of the Morris type, constituted of two fine * It may be remarked that without such protection, the effect of any atmospheric disturbance upon the indication of the hot-wire anemometer employed is under certain conditions considerably magnified by the operation of the jet. A combination of jet and hot wire, such as described herein, would appear to be an exceedingly sensitive device for indicating small atmospheric movements. Orijices, and Entrainment of Air by the Issuing Jet. 971 platinum wires. The end of this tube away from the jet opened into a large vessel containing water, which served to saturate the air with water vapour, a condition corresponding to that of the air employed in calibrating the instrument, and to shield the anemometer from outside disturbances. Slightly different calibration curves were employed corresponding to different atmospheric temperatures. The velocity with which the air was set into motion in the tube E by the jet depended upon the length of tube employed. This was chosen of the minimum length consistent with securing steady readings of the anemometer, and was equal to 54 inches. The jacketted tube A was inserted into the tube D to such a depth that the velocity of air-stream passing over the anemometer wires was a maximum for the pressure employed. A gas-tight joint was then made between the tubes A and D bv means of the gland G screwing down on a rubber ring. If for any cause it became necessary to remove the tube A, it could be readjusted to its former position by reading the position of the pointer P with reference to a scale marked on the tube. The pointer S moving over a circular scale served to indicate the azimuthal position of the orifice disk, and was more especially used in connexion with subsequent experi- ments with multiple-orifice disks. (ec) Measurement of the Discharge.—The method employed for determining the rate of efflux of air was to determine the time taken for a definite measured volume of air to flow through the orifice. The device employed for this purpose is shown in fig. 2, andisa slight modification of the apparatus employed by Coste*. The measured volume of air is contained between two marks shown on the vessel A. Water is delivered to the vessel through the jet B from a Marriotte bottle or overflow constant-head device, the rate of flow of water being controlled by means of the cock C. The cock D is fully open during the flow of water through B, and closed while the vessel A is subsequently emptied through the cock H, the cock © being meanwhile unaltered. In this manner the rate of flow of water through B is very conveni- ently adjusted to the series of increasing rates of flow employed in the present sequence of experiments. Vessels A of various sizes were employed in the course of the present experiments, the smallest having a volume between marks of 8382 c.c. and the largest a volume of 7081 ¢e.c. The lower stem of the vessel A extended for a eonsiderable distance below the lower mark, so that the * J. Soc. Chem. Ind. vol. xxx. p. 258 (1911), dR 2 972 Dr. Thomas on Discharge of Air through Small conditions of flow became steady before the time of the _ water surface crossing the lower mark was taken. In some of the larger vessels a small bulb was blown on the lower limb for the same purpose. The pressure in the vessel A was measured by a water manometer connected to F, and the temperature was measured by a thermometer hanging near A. Air was delivered to the orifice through the tube G, and passed over calcium chloride and then through a small plug of glass wool. Throughout, the volume of air delivered was corrected for pressure, temperature, and humidity. The tube H served for drawing air into A, the cock K being meanwhile closed. The device maintained a very constant pressure at the orifice, any variation occurring being somewhat less than of - the order of 0°5 per cent. of the total pressure. ; (d) Preparation of the Orifices.—Considerable attention was given to the preparation of the orifices. Throughout, a Orijices, and Entrainment of Air by the Issuing Jet. 973 tlle endeavour was made to make the orifices as circular and smooth as possible. In the case of the orifices numbered 1- 10 in the sequel, these were made in a stiff copper-nickel alloy (88 per cent. Ni, 12 per cent. Cu), 0°0229 cm. thick, and except for the smaller ones were made by means of a machine designed by Dr. Charles Carpenter for the bulk manufacture of single or multiple-orificed injectors for use in gas-burners. By this machine, a disk about 6 mm. in diameter is cut froma sheet of metal, aud simultaneously a hole or holes punched in the disk by the passage of a flat- ended accurately cylindrical needle or needles through the disk, which is held meanwhile between blocks, the lower carrying the needle or needles, which after passing through the disk enter accurately bored holes in the upper block. The machine is operated after the manner of an embossing press. Orifices in disks prepared in this manner possessed a very smooth interior surface, and there was little burr on their outer surfaces. This was readily ground away by rubbing on an oiled stone, and the inner surface finally burnished by means of a cylindrical needle. The two smaller orifices employed were drilled in the same material, great care being taken to see that they were as accurately circular, and their interior surfaces as smooth as possible. Fig. 3 (Pl. VI.) shows microphotographs (linear magni- fication about 37) of representative orifices of the series, They indicate very slight departure from the cireular form. Actual measurement showed that the greatest and least radii of any disk agreed to within 1 per cent. except in the case of No.8, where a maximum difference of about 2 per cent. occurred. A microscopic examination of the interior of the orifices showed that there was present very little roughness in the finished orifices. The prepared disk was carefully soldered on to a cap which screwed on to the lower end of the tube B (fig. 1), this cap being then itself soldered on to the tube. It was previously ascertained that all joints in the various parts of the apparatus were gas-tight. Tested under 10 inches of water pressure at various times during the course of the experiments, the leak in the discharge tube when the orificed disk was replaced by a blank of the same thickness, and in the anemometer tube, was found not to exceed 0°001 cubic feet per hour. 974, Dr. Thomas on Discharge of Air through Small RESULTS AND DISCUSSION. Table I indicates the general nature of the observations and the calculations based thereon. (a) Discharge.—Fig. 4 shows how the discharge through the various oritices of the series depends upon the pressure and diameter of the orifice. The results shown in this figure all refer to orifices in disks’ of thickness 0°0229 cm. f "DIAMETER OF ORFICE (CM) | |NCl Gy4621, N°6 A 0870 | IN°2 ©1868, N°777-060 N°3 X20, N°8 G-0442 N°4 4049, N°9 D 0335 25 © 0942, NO V 0256 LOG OF DISCHARGE IN CUBIC FEET PER HOUR. LOG OF PRESSURE IN CMS OF WATER Particulars of the diameters of the various disks are given in the diagram and are tabulated in Table II. below. In the figure, the logarithms of the discharge are plotted as ordinate against the logarithms of the corresponding pressures as abscissee, all volumes being reduced to 0° C. and 760 mm. pressure, dry. It is clearly seen from fig. 4 that the experimental results may be represented algebraically with considerable accuracy by a linear relation between the logarithms of the pressures and discharges respectively. The lines drawn in fig. 4 are the “ best fitting ” straight lines which can be drawn through g Jet. SUN of Air by the Ts ‘ainment o ces, and Entrai tf Or TABLE I, Orifice No. 5. Thickness of Disk, 0:0229 cm. Diameter of Orifice, 0:0942 em. =. b- .) {fine ana oo Se ee eee | Corrected as discharge | Vol. of air Temperature. Tal Tinie f through Anemometer. | entrained | Vol. of air Pressure (0° C.) A Ee dischatee orifice. : by jet. entrained a _Barometer.| marks on of ont Deflexi Nat ; by rere Sere! | | (inches) bottle | measured | P®*20" | Galyano- ; emi wetness eT [eer ee (cms. of | | anitaigtedel Moclanne dry, meter | Balance | Bridge | right—left., measured | issuing water) | Bottle, | Water | ( - : (secs.) "| measured | 974 | tesistance.| current. (mm, ) at 0° C. | through | "| jacket. | rade peo at 0° C, (chms) (ohms (amp.) and orifice. and 760 mm.) | | 760 mm.) 0°320 | a7 3), 12%) 30°45 2361 661 0435 10 1014 1:000 (2 1°56 3°59 0502 | 12:5 | 13-0 542 0533 10 7 sj 110 1:98 371 0°850 130 | 130 | All 0701 10 A o 258 2°81 4:01 1348 | 13:3 13:0 | 325 0:886 10 % ” 394 3°80 4:29 1°664 | 15°5 13:0 | 295 0974 10 ‘A 7" 466 4:46 4:58 P80: a1 13° 13°5 275 1:043 6 5 ” 314 5°30 5°08 265 | 130 | 13:5 239 1199 6 7 A 379 6°42 5°35} JOD. \auleo | 130 201 1-428 6 %9 ” 448 8:99 6:28 £88. i isp |; 125 |. 30:46 175 1624 6 ” ” 303 =| «11:20 6°90 608 | 137 | 125 160°2 1771 2 1028 1:200 335 13°58 7:67 9°58 | ae 128°2 2°22] 2 1025 5 382 18:46 8°31 1}*44 | 12°5 | 4416 2188 2°432 2 ” ” 400 20°64 8:49 15°10 | ha 10:0) | 194°5 2°746 2 ai | 5 182 | 145 | 130 | 1776 3°015 2 sia hoe. as 204 | 1 130: | 168°2 3'184 2 3; ) 200) 445), 130 | 30°44 153°0 3518 2 i | ‘5 | | 976 Dr. Thomas on Discharge of Air through Small the several points. Hach has been drawn through the ey) “‘centre of gravity” — , = of the respective observations, 1 at an inclination @ to the axis of logarithms of pressure given by tan ed =), ee) 2(a—%)? where Ra eae = a0, «x and y being the logarithmic ee experimentally determined and m the number ofobservations. It follows that the dependence of the discharge Q (measured at 0° C. and 760 mm. pressure) upon the excess pressure eé can, within the limits of pressure employed in the present series of experiments, be represented by a relation of the form QaNe. | a The respective values of A’ and «@ are set out later in Table II. It is cf interest here to consider the relation of this equation to that deduced for the relation of discharge to pressure on the assumption that the discharge occurs under adiabatic conditions. Lamb * gives for the mass discharge under these con- ditions the formula ytl 2 \12 n\y )te, ie nas’ = (7) nly GS) Sy ay where p, and po are the respective pressures outside and inside the vessel from which the discharge occurs. pp and ¢% are respectively the density of the gas and the velocity .of sound inside the vessel, S’ is the area of the vena contracta, y the ratio of the 10 Diameter. (em.) 01621 01368 01201 0°1049 00942 0:0870 00607 0°0442 0°0335 0:0256 002064 0:01469 0°01132 0008644 0006971 0005944 (002894 0:001534 0000881 0000514 ) | | TaBueE II. Thickness of Disks, 0°0229. —SS Empirical Forinula for discharge in ¢.c, per sec. Corresponding best value of « Winlineroninl measured at 0° tS : (sq. em.) and 760 mm. S’(2ep,)? fre 2 A’e™, Po. ( pia NG Gre) 36) 0:01327 | 11:79 6482 3:0 | 0:0097 1 DiB6) wer qo 27 Ee 000772 ‘ Pe 7-12 6486 | 2:3 = 0:00588 5:94 ¢0'480 3:3 0:00488 AGS oy 25 J 000407 DBA eee = 0:8 0002113 epee! Om ans Br) ‘! 0-001101 O786e0°= Re | 0:000628 Ogee. ~106 | 0:000320 Coefficient of Contraction. S' 3° 0-679 0-701 0-684 J 0 730 0°718 0713 0622 980 Dr. Thomas on Discharge of Air through Small ordinates and abscissee respectively. The relation between the logarithms in the case of the larger orifices is clearly linear, the points corresponding to orifices nos. 1-7 lying very accurately on the best-fitting line given by Xe) O00) eee For orifices nos. 8-10, the value of the index increases as the diameter of the orifice diminishes. This result, together with the very approximate constancy of the index a for disks nos. 1-6 shown in column 4 of the table, indicates that these disks only of the series employed can be regarded as thin disks. Disks 8, 9, and 10 are to be regarded rather as orifices in thick disks, disk no. 7 affording a transition from one class to the other. The results indicate that an orificed disk is to be regarded as thin if the diameter of the orifice is not less than about three times the thickness of the plate. In the recent paper of Buckingham and Edwards * on the efflux of gases through small orifices, the diameters of the orifices employed were considerably smaller than any employed in the present work, and moreover, were in no case greater than 1°7 times the thickness of the disk, being in three cases out of four very much less than this. Such disks would, in the light of the present work, be characterized as thick disks. The average value of the contraction coefficient in the present work was found to be 0°674. For orifices of diameter d cm. in then disks (nos. 1-6) of thickness 0°0229 cm., the discharge of air is given by the formule Vol = BOQ dre e0eee0008 aoe Se Nae 1/2 = 0674s CO (1-2-9 2 , eae the symbols having the significance given in the text. Of these, formula (vi.) is of a type which has some physical justification. In addition, (vil.) is correct from the point of view of physical dimensions. For purposes of comparison between orifices in thin and thick ‘* disks,” experiments were in like manner carried out with very carefully prepared short channels bored in brass. Particulars of the various channels are given in Table III., together with empirical formule, the detailed results being plotted in fig. 5 and the empirical formule deduced as already explained. * Scientific Papers, Bureau of Standards, vol. xv. p. 584 (1919-20). LOG OF DISCHARGE IN CUBIC FEET PER HOUR Orifices, and Entrainment of Air by the Issuing Jet. TABLE III. ISL Length. (em.) channel, (cm.,) 0:0780 0:0612 0:0544 0:0471 0:0421 \ 0°0421 00424 0:0420 0:0442 \ | | ] | | Empirical Formula | for discharge (A’e® Diameter of | ee) (measured in c.c. per sec. at 0° C, and 76 mm.) at excess pressure € cms, of water, 3:39 0919 2-09 "880 1°65 go 085 1:05 @264 0-783 ¢0°628 0:509 6721 0:479 60783 1:19 0509 1:32 60007 982. Dr. Thomas on Discharge of Air through Small For the channels 1 A-5 A of constant length 0:2789 cm., the value of A’ in the empirical formula for the discharge A’ e is given in terms of the diameter d by the linear relation A'=73d — 2°33. The gradual increase of the index @ in the case of channels 1 A-5 A as the diameter of the channei decreases is seen from fig. 5. It will be noticed that at a pressure a little greater than the maximum employed in the construction of fig. 5, the straight lines shown corresponding to the orifices 11 and 5A of the same diameter intersect (actually the pressure was found to be 33°2 cm. of water). For pressures greater than this, the discharge through the longer channel is actually greater than that through the shorter channel of the same diameter. A similar phenomenon is represented by the points P and Q, in which the straight lines corresponding to channels 8 and 11 in disks of thick- ness ('0229 cut the line corresponding to channel 4a, of slightly larger diameter and of about 12 times the length. The phenomenon is clearly attributable to the difference in the form of the issuing jet in the respective cases of discharge through a channel in a thin or thick plate, the existence of the vena contracta in the former case reducing the effective area of the discharge and tending to counterbalance the effect of the greater length of the channel in the latter. Attention has been directed to the existence of a critical length of channel, such that the discharge through an orifice of given size Is a Maximum, In a recent publication of the Burean of Standards, Washington *. (b) Air Entrainment.—In fig. 6, the respective total volumes of air (reduced to 0° C. and 760 mm. pressure, dry) entrained by the issuing jet in the case of each of the jets nos. 1-10, are plotted as ordinates against the respective pressures as abscissee. The several curves are numbered according to the number of the corresponding orifice. Curves 1-7 represent the normal manner in which the volume of air entrained by a jet of air issuing from a given orifice increases as the pressure at the orifice is increased. As the pressure is increased, an initial approximately linear increase of the total volume entrained is followed by a subsequent increase at a continuously decreasing rate, the curve becoming concave to the axis of pressure. ‘This latter is to be anticipated, as the viscous and frictional drag upon the stream of entrained air increases as the velocity of the stream increases. The curves corresponding to orifices nos. 8, 9, and 10, * Technologic Papers No. 193, p. 17 (Sept. 6, 1921). Orifices, and Entrainment of Air by the Issuing Jet. 983 which, as has been shown, must be considered essentially orifices in thick disks, exhibit points of inflexion at R, Q, and P respectively corresponding to pressures of 3, 5, and 12 em. of water. During the course of the experiments with orifices 9 and 10, anomalous behaviour of the jet in the region of these respective pressures was very readily detected, as the galvanometer reading became very unsteady unless the jet was protected very carefully from outside disturbances. Steady deflexions could be obtained by. reducing outside atmospheric disturbances to a minimum. 24 (CUB FT. PER HR.) VOLUME OF AIR ENTRAINED BY JET 10 15 PRESSURE (CMS. OF WATER) For values of the pressure at the orifice, below the critical value, the galvanometer deflexions were exceedingly steady. The galvanometer deflexion was unsteady until the value of the pressure was increased to a definite value, above the eritical value, and thereafter the deflexions were again extremely steady. The phenomenon indicates the existence of a small range of pressures over which the issuing jet is essentially unstable, the efficiency of the jet as regards air entrainment being very considerably affected by slight disturbances in the surrounding atmosphere. The nature of the instability of the jet is most clearly 984 Dr. Thomas on Discharge of Air through Small seen from a consideration of fig. 7, in which the volumes of air entrained per unit volume of air in the issuing jet are plotted as ordinates against the pressures as abscisse, in the case of orifices nos. 1-10. Curves 1 to 8 inclusive indicate the normal behaviour of the jet as regards air-entrainment. It is seen that a diminution in the size of the orifice is accompanied in each of these cases by a greater air-entrain- ment per unit volume of air in the jet. This point is of Ib wn a . VOLUMES OF AIR ENTRAINED PER UNIT VOLUME OF AIR ISSUING FROM JET. 56. i=) PRESSURE (CMS. OF WATER). importance in the design of gas-burners operating on the Bunsen principle, as it is to be anticipated that a greater degree of primary aeration of the issuing Jet of gas can be effected by diminishing the size of the ejector orifice, and employing a multiple-orificed ejector in place of an ejector provided with a single orifice of greater area. In the practical application of this result, however, the several Orijices, and Entrainment of Air by the Issuing Jet. 985 orifices must be so disposed that the air-entraining power of each individual jet is not reduced by mutual interference of the jets. It will be clear from the figure that there is a limit of size below which the orifice cannot be reduced without the intro- duction of disturbing factors adversely affecting the greater degree of air-entrainment normally accompanying reduction of the orifice. This is first seen in the region of F in curve 9, where the degree of air-entrainment is somewhat less than its anticipated value, as shown by the broken line. From curves 9 and 10 it is seen that over the range of pressures up to about 16 cm. of water, the degree of air-entrainment effected by the jet issuing from the small orifice (no. 10) is very much less than that effected by the jet issuing from the larger orifice (no. 9). A similar phenomenon to that described has been observed in the case of coal-gas issuing from a fine orifice. In such a case a flame burning at an orifice under conditions corre- sponding to those shown at Fin fig. 7 would be very sensitive to slight changes of pressure, the degree of air-entrainment varying considerably with a slight increase of pressure. The phenomenon probably explains, at least in part, the existence of low-pressure sensitive flames, to which attention has been recently directed, and which were studied in considerable detail by Chichester Bell *. With a view to ascertaining the cause of the apparently anomalous bel aviour of jets issuing from orifices 9 and 10, similar experiments on air-entrainment were performed with air jets issuing from the comparatively much longer channels nos. 1 A-5 A, 5B, and 5C, particulars of which have been given in fig. 5 and Table III. The curves corresponding to those in fig. 6, in which the total volumes of air-entrainment were plotted as ordinate against the pressures as abscissa showed well-marked points of inflexion in the cases of channels 3 A,4A,5A,5 B, and 5 C, these points corresponding to pressures respectively equal to 2°5, 5:0, 7°5, 11:0, and 13 cm. of water. The curves for channels nos. 1 A-5 A showing the degree of air-entrainment per unit volume of air in the jet are shown in fig. 8. The curves generally resemble those given in fig. 7. One distinction is of importance. A comparison of curves 3A and 4A with curves 9 and 10 in fig. 7 indicates * See, e. g., ‘ Nature,’ vol. cviii. p. 532 (1921). Phil. Trans. Part 2, pp. 383-422 (1886). Phul. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 38 386 Dr. Thomas on Discharge of Air through Small that the effect referred to is not so pronounced in the case of jets issuing from the longer channels. The same is seen from curves 4A and 5A in fig. 8. The existence of turbulence in the jet naturally suggests itself as the cause of the phenomenon referred to. It is clear that as the effect occurs only over a limited range of pressures, and is not present at the higher pressurés employed in the present series of experiments, and as moreover the mean VOLUMES OF AIR ENTRAINED PER UNIT VOLUME OF AIR ISSUING FROM JET fo) Shan lo esis ee 25 PRESSURE (CMS OF WATER). velocity in the jet for this range of pressures is considerably, below the critical velocity at which turbulent flow occurs, the turbulence referred to is impressed upon the jet on entering the orifice. It is therefore to be anticipated that the effect would be less pronounced in the case of long channels than in the case of small channels, as in the former case the unstable turbulence initiaily impressed upon the stream would die away to a greater extent than in the latter case. This, as has just been stated, was found to be Orijices, and Entrainment of Air by the Issuing Jet. 987 the case and is clearly brought out in fig. 9, which gives the results obtained with channels all of the same diameter but of different lengths. The effect is much less marked in the case of the longer channel 5C than in the case of the shorter channel 5 B or 5A. The absence of the effect at the lower pressures is attributable to motion of the gas at such low pressures being IS - : is ORIFICE ssc | (cm) 3 5A 0:2789 0 : NY VOLUMES OF AIR ENTRAINED PER UNIT VOLUME OF AIR ISSUING FROM VET PRESSURE (CMS OF WATER) in general so slow that little initial unstable turbulence is impressed upon the stream, and any such turbulence, if produced, is damped out before emergence of the jet from the orifice. At the higher range pressures at which the effect is absent, the passage of the air into the channel approximates to stream-line motion, the air within the reservoir immediately adjacent to the disk being practically stagnant, so that little turbulence is produced by a sudden 882 988 Dr. J. R. Partington on the change of direction of motion of the air entering the channel or orifice. The work detailed above was carried out in the Physical Laboratory of the South Metropolitan Gas Company ; and the author desires to express his sincere thanks to Dr. Charles Carpenter, C.B.H., for the provision of facilities for carrying out the investigations, and for his continued interest in the work. 709 Old Kent Road, S.E. 15. 14 Aug., 1922. LXXXVII. Zhe Chemical Oonstants of some Diatomic Gases. By J. R. Partineton, D.Sc.* ile fee Theorem of Nernst supplements the two Laws of - # Classical Thermodynamics by assigning to the constant of integration, I, of the Reaction Isochore of Van’t Hoff (1) a value otherwise determinable only by experiment with the particular system of materials under consideration : dlog K/dU=Q,/RI? .. . eel loo, K= —QURI+L, . > ee According to Nernst, l=2ni, . 1. 9) es i. e. the integration constant is represented as the algebraic sum of the products of the numbers of interacting molecules (n) by the chemical constants (1) of the various pure gaseous materials taking part in the reaction. 2. The value of 7 is the constant of integration of the Clapeyron-Clausius equation, simplified by the usual approxi- mations Agq=RIP.dlog pd, . <2 ees) where Xp is the latent heat of vaporization per gram molecule at the temperature T. The value of A, as a function of temperature is given by Kirchhoft’s equation My=ry+ | (Coal ee | (SS where C, is the molecular heat of the vapour at constant pressure, and ¢ that of the condensed phase. Thence r etal loger=— Rt RI, al (C,—c)dT +7. . (6} * Communicated by the Author. Chemical Constants of some Diatomic Gases. 989 According to Langen*, the value of C, may be split into a translational term independent of T and equal to 5/2.R for a diatomic gas, and a rotational term, C,,, dependent on T’. Equation (6) is then written r 5 LW Geecer bot Coe Saag play log. p= — RT + slog. T + zl. | (Cp-—c)dT +2, (6a) log p (atm.) = — a +2°5 log T Um tai he Fare art al (Cp —e)dT+C. . (68) The constants i, C then supply the terms to (3), and permit the integration of (1). The values of © have been calculated from a consideration of the experimental data by Langen for several gases. Values of C had been calculated by Nernst f by a somewhat arbitrary method, and although his results provide a satisfactory approximation for particular problems, they are not in accordance with modern theories of specific heats, as was pointed out by the author in 1913 f. 3. It will be noted that the equations so far given do not provide a complete solution of the problem of predicting, from purely thermal magnitudes (heats of formation and specific heats) and universal constants, the behaviour of materials when placed together under specified conditions in the absence of passive resistances §. The final step was taken in the case of monatomic substances by Sackur and by Tetrode ||, who were able to calculate the value of 7 in terms of universal constants. If the temperature is reduced to such an extent that the atomic heat of the condensed phase becomes negligible in comparison with unity, then (6) becomes Xo logep= — pn +2'5 loge T +2, pi COE) since C,, is zero for a monatomic substance, and the value * A. Langen, Z. Elektrochem. xxv. p. 25 (1919). . + ‘Recent Applications of Thermodynamics to Chemistry,’ 1907. Theoretische Chenne, 8-10 Aufi., 1921, p. 799. { ‘ Thermodynamics,’ 1913, p. 496. § It may be that the influence of passive resistances could be included by introducing a ‘‘heat of activation,” in the sense understood by Perrin, Trautz, W. C. M. Lewis, and others, in connexion with dA, but this problem is not considered in the present communication. —_- ” || Sackur, Ann. d. Physik, xl. p. 67 (1913). Tetrode, bed. xxxyiii. », 434, xxxix. p. 255 (1912). Stern, Z. Elektrochem. xxv. p. 66 (1919). 990 Dr. J. R. Partington on the of 7 is then given, according to these authors, by 3/2 75/2 i=log.@™™),* oe 4 where m is the mass of the atom, kis Boltzmann’s constant (R/No, where N, is Avogadro’s constant), and / is Planck’s constant. With numerical values (see § 7 below), and p in _atm., this gives J=—1-58941-5lopM, . . Ae ie where M is the atomic weight, referred to the same standards as No. : 4. The object of the present communication is the extension of this line of investigation to a hypothetical diatomic molecule which, 1t is believed, represents with some approxi- mation the structure of a particular group of gases*. A general solution would obviously enable us to predict the results of all types of gaseous reactions without recourse to experiment, and would provide a long-sought solution to a fundamental problem of chemical affinity. The method of calculation adopted is that of generalized statistical mechanics f. An isolated system, possessing an energy e, and composed of a large number of molecules which exert no forces on one another, is assumed to be definable in terms of a set of generalized coordinates G1) Ya) -»-») and a corresponding set of generalized momenta 1, Pa --. related by the first canonical equation of Hamilton, Oh = 06/071 . : Me Peo OS (8) According to the Quantum Theory, | H=((.. dq, dgo...dp, dpe ..., which is independent of time and of the particular choice of coordinates, has a definite value for each element of the generalized space (Llementargebiet). In the case of an ideal gas, the mean energy € coincides with the energy € in any point of the element, and G Sec al) ¢ Bay * Partington, Trans. Faraday Soc. 1922. | J. W. Gibbs, ‘Elementary Principles of Statistical Mechanics.’ Planck, Warmestrahlung, 4 Aufl. 1921. Jeans, ‘Dynamical Theory of Gases,’ Chemical Constants of some Diatomic Gases. 991 5. In the first case we suppose the gas molecule to consist of two identical atoms rigidly ‘attached to each other ata fixed distance. In addition to the coordinates of the centre of gravity we require two angles, @ and ¢, defining the direction of the molecular axis. Rotation about this axis is, as usual, ignored. We then have . N=2®3 H=Y3 B=2Z3 U=F3 B=: Pi=ML 3 Pp=MY 3 pz=MzZ; py=MK7O; ps =mK?’ sin’ Od, where K is the radius of gyration. The energy of this molecule is given by a pop +e) 4 js (62+ sin? 062) +e, . (10) e= 5 (i where € is the energy of i molecule at rest in the generalized space. Thus aoe 5K4 (it cal eens Bema” \\ Mena nld: dda dbdde (1) The limits of the multiple integral are the boundaries of the element of volume for a, y, and z; the angles 0 and 27 for @; the angles 0 and w/2 for ¢, and all the velocities from —2% to +0. Hence, if V is the total volume, eae Ke Pipe a sca a Le a The free energy, w, of the system is then given by * Ap — —kTN { log. Le - i aaa log. Dash (12) where N is the iotal number of molecules in the volume V, say N=N,, the number of molecules in one gram-molecule. In the above case Ene (QarmkT)*?e **, = KING log (QrmkT)? + Noe. . (13) For equilibrium between the vapour and the condensed phase Pl Apeap( Vee Vi ee 8) et CQ in which dashed symbols refer to the condensate. Substi- tuting in (14) from (13), and neglecting small terms, we find 2Qar KK? Ve 5/2 Chic ap" INg erg f k log. 873 —(2arkmT)* Sr ane ie. * Planck, loc. cit. p. 210. 992 Dr. J. R. Partington on the But e9—wW'/No=rAy’, the latent heat of vaporization per molecule at T=0; hence, with the substitutions pV =N kT = 5 oe ; and Ay>=NoAy, we find K?(Qark)"” CSP am where H=A’ in the peueeaieed space of five dimensions *. log. p= — ah 5 loge’ Uae 2 9 Oke M+ log, 6. For a diatomic gas of the type considered in § 5, C,=7/2, and hence log p(atm.) = mee ee e717 +C, log T+2°5 log M +2log K+12°730. . (15a) (hee 137 x 10-% 5 k= 655 X 10-7 = No = Onn ame iatms—1013250 abe tonite. Gee Millikan, Phil. Mag. July TEA The equation representing the vapour-pressure of a diatomic substance at such low temperatures that the energy of the condensate is negligible in comparison with that of the vapour (which will generally occur before the gas begins to lose its diatomic character, except in the case of molecules of very small mass and diameter, such as hydrogen) is los p= we + Oplog, T +7, or log p(atm.) = be71t t+ Cy log T+ C. res i By comparison of (15) with these we find ei. (AN?2)° 2. 7G or C=2°5 log M+ 2 log K+ 12-730. 7. In the case considered, K?=7?, where 7” is the radius of the molecule. For oxygen, r=1'8x10-? cm.t, M=32, hence Co,=1°001. The four values given by Langen (loc. cit.) range from 0-539 to 1:021, the mean being 0°829. In the case of nitrogen, r=1:9X 1078 em., M=28, hence Cy,=0°904. Langen gives only one value for nitrogen, —0-05, from which one can perhaps only conclude that it is somewhat less than the value for oxygen. The case of i=2°5 log. M+2 log. K + log, ——{— * The various methods of quantizing rotations are kept in mind. + Jeans, ‘ Dynamical Theory of Gases,’ 2nd edit. p. 341 ; all values of 7 from this. Chemical Constants of some Diatomic Gases. 993 hydrogen is probably not satisfactorily covered by formula (16), since the value of Cy becomes appreciably reduced within a region of temperature before the value of c for the condensate becomes negligibly small. Hydrogen, therefore, should behave in a manner intermediate between that of a diatomic gas (equation 16) and that of a monatomic gas (equation 7). Nernst * has applied (7) to hydrogen, and after the application of small corrections, has found Cy, = — 1°23; whereas Langen, by formula (6), finds —3°767. Hguation (16) gives Cy,=—2°255 (M=2°016; r=1°34 x60 cm.). 8. In the case of gas molecules composed of two different atoms rigidly bound together, the calculation is similar, except that the angle ¢ is now, since the molecule is no longer symmetrical, to be taken between the limits 0 and 7. In the cases to be considered it is still a sufficient approxi- mation to take K?=7r*. This case is, therefore, covered by the addition of log2 to (16). In the case of carbon monoxide, r=1°90 x 107°, M=28; hence Coeg=1°205, whilst Langen gives —0°04. [for nitric oxide, »=1°86 x 107%, M = 30; hence Ono = 1:268, whilst Langen finds 0 92. Perhaps all that can be definitely said of Langen’s values for these gases is that they are somewhat larger than the vaiue for nitrogen, and it is noteworthy that Nernst’s empirical values for the compound gases are larger than those for the elementary gases : 3°5 for CO and 3°5 for NO, as compared with 2°8 for O, and 2°6 for Nf. It is believed that the above method of calculation gives results which are in all cases of the right order, and that the values deduced by other methods are still so divergent that a more searching comparison is not at present possible. It is hoped that the method will shortly be extended to gases with more complex molecules, in which internal motions also occur. If these are considered as small vibrations, their energy can be represented as the sum of squared terms in the coordinates, and the above method can be applied to them without difficulty. East London College, University of London. t Grundlagen des Neuen Wéarmesatzes, 1918, p. 150. There are a few misprints in this section, e.g. in (120) —0°5T should be —0‘d/nT, and (2mm)? should be (24m)*’. * Theoretische Chemie, p. 799 (1921). 904. | LXXXVIII. The Motion of Electrons in Carbon Diowide. By M. F. SxinKER, Rhodes Scholar, Exeter College, Oxford *. 1 some recent publications of the Philosophical Magazine, Prof. J. S. Townsend and Mr. V. A. Bailey t describe their experiments on the motion of electrons in hydrogen, nitrogen, oxygen, and argon. In this paper I wish to give the results of similar experi- ments with carbon dioxide and to compare the results. The apparatus used had the same dimensions and was similar to the one described in the above papers. The electrode E,, in fig. 1{f, however, was not exactly under the slit in B, but was 0°6 millimetre to the right. In order to find the velocity of agitation u it is necessary to find the normal distribution-curve when the centre of the stream is 0°6 millimetre from the centre of the electrode H,. In this case R, the ratio of the current received by the central electrode to the total current, is given by the curve in fig. 2, Z being the electric force in volts per centimetre. | The curve differs slightly from the curve which corre- sponds to the case in which the centre of the stream coincides with the centre of the electrode Hp. * Communicated by Prof. J. S. Townsend, F.R.S. + Phil. Mag. vol. xlii. Dec. 1921, and vol. xliii. March 1922. { Fig. 1, vol. xli. p. 875. Motion of Electrons in Carbon Mowide. 995 The calculation of this curve will be explained in a future paper by Prof. J. 8. Townsend and Mr. V. A. Bailey. In order to find the velocity of the electrons in the direc- tion of the electric force, two different magnetic forces may be used. With this eccentricity of 0°6 millimetre the stream may be deflected 1:9 millimetres to the left or 3:1 millimetres tothe right. In these experiments all deflexions were to the right, as the determinations with the larger deflexions are the more accurate. The results of the experiment are given in Table I., where p is the pressure in millimetres of mercury, k the factor by which the kinetic energy of the electron exceeds the kinetic energy of a molecule of a gas at is’ ©... W the velocity of the electrons in the direction of the electric force in centimetres per second. TABLE I. p Z Z/p k. WwW x107°. 20°23 416 0:206 1:19 _ 9°82 2:08 0:222 1:283 1:18 9°82 4:16 0:444 1:29 2°41 5:06 2:08 0°411 1-277 2°45 9°82 8:33 0°888 — 4°91 506i 4°16 0:822 1°36 4°55 2°49 2°08 0°835 1:36 4°67 5°06 8°33 1:647 ~ 9:42 2°49 4°16 ays 1°72 981 1:26 2:08 1:66 1°64 9°47 5:06 16°67 3°30 -— 22°4 2°49 8°33 3°32 2°88 23'8 1°26 4°16 3°29 2°79 23°6 62 2°08 3°32 2°89 24°5 2°49 166 6°64 221 82°4 1:26 8°33 6°59 21-1 82°4 63 416 6°64 23°1 81°4 2°49 go'oo 13°4 60°6 118 1°26 16°67 132 60°1 124 1:26 33°33 26°4 81°3 142 63 16°67 26°6 91 150 63 33°33 53°2 147 202 ee OO — ree 996 : Mr. M. F. Skinker on the §7.The values of W and £ are plotted against 2 in figs. 3-6, together with the curves for hydrogen and nitrogen. Fig. 3. loa cal mapa) 997 Carbon Dioxide. Motion of Electrons in 998 Motion of Electrons in Carbon Dioxide. In the following table, u is the velocity of agitation of electrons in centimetres per second, / the mean free path of the electron in centimetres, p the pressure of the gas in millimetres of mercury, and ® the proportion of energy of the electron lost in collision with a molecule. The formule connecting u, /, and X with the quantities k and W being :— u=1:15x10'x Wk, We oe 1 Paste WX = 2°46 x ae TABLE II. Zipaterce Neg: wxi07® uxi0s®. mxied. Meco. 50 139 19°5- 185°7 3°67 506 4() 117:5 17°75 1248 3:64 > | Age 30 96 159 1128 4:15 487 20 75 13°8 99:5 4°76 472 10 47 10°8 73:9 591 ~—- 460 6-5 20°7 Hise a 524 4:36 543 5:0 9 5:0 34:5 2:39 516 4 4:8 3:2 25:2 1:40 397 3 2:3 2-0 17°5 ‘809 321 9 18 2 Fs 15°4 ‘630 144 1 15 “5D 141 538 37:4 O°5 1:3 ‘25 13°1 454 8:95 0:25 12 12 12°6 ‘419 2°34 In order to determine whether or not there were any ions | in the stream, the magnetic force was increased, to see if the stream were completely deflected off E, and H,*. This was found to be possible when using a magnetic force which was comparatively small and which would not have been sufficient to deflect ions from the plates. Also the quantity & and the velocity W were found to remain constant with different Z values of Z and p when — was constant ; these results show that there could not have been any permanent ions formed in the gas. * Fig. 1, vol. xlii. p. 875. A Wide Angle Lens for Cloud Recording. 999 With values of : greater than 30 the loss of energy in a collision is comparatively large, so that the velocity of agita- tion is less than seven times W, and in these cases the formula for W in terms of / and u is not so accurate as in the cases r where a is less than 30 and wu comparatively large. p "G a 7, Table II. shows for the higher values of 2 that the mean free path increases with decrease of u, but for the lower values it decreases with decrease of velocity of agitation. In the other gases the mean free path increases for the smaller values of the velocity of agitation. The values of X% show that with this gas there is a remarkable increase in the loss of energy of an electron in a collision for comparatively small increases in the velocity of agitation from the values 13 x 10’ centimetres to 15 x 107 centimetres per second. Electrical Laboratory, Oxford, July 1922. LXXXIX. A Wide Angle Lens for Cioud Recording. By W.N. Bonn, W.Sc.(Lond.), A.R.C.S., AJdnst.P., Lecturer in Physics, University College, Reading*. [Plate VIL] HIS paper consists of a short description of a lens that might be used for obtaining a photographic record of the clouds visible at a meteorological station at definite times, or for similar purposes, such as recording lightning flushes. The special feature of the lens is that its field of view embraces a complete hemisphere; so that if the lens be arranged to face vertically upwards, all the clouds visible at the station at any one time can be recorded photo- graphically on a single flat plate or film. The resultant photograph (see Pl. VII.) is circular, any clouds at the zenith being reproduced in the centre of the circle, and any near the horizon appearing near the edge of the circle. Such apparatus might, of course, be used at two stations simultaneously to obtain the altitudes of the clouds. * Communieated by the Author. 1000 Mr. W. N. Bond on a The lens in its original form consists of a glass hemi- sphere L (fig. 1) of radius vr. The light is incident on the plane face, which is covered by a thin screen S, except for a small circular aperture at the centre. It will be seen that a ray incident in the plane of the outer face of the lens will be refracted at the critical angle. Furthermore, all refracted rays will arrive almost normally at the hemi- spherical face of the lens, and will subsequently converge to form an image, which, for objects at infinity, will lie on part of a sphere III, concentric with the hemispherical face of the lens and of radius ry/(#—1), where yp is the refrac- tive index of the lens. The emergent cone of rays will in the aperture in the screen S is small enough, the whole image will be sufficiently in focus on a flat plate PP placed at a distance from the plane face of the lens equal to the mean distance of the various portions of the true image III from this plane face. The screen 8 should be covered on the outer side by a plane plate of glass G, the whole being cemented together. This arrangement avoids the finite thickness of the screen 8, preventing some or all of the light incident at fairly oblique angles from entering the lens. ) It will be seen that the photographic plate should be placed at a distance h from the plane face of the lens of about 2°57 (i.e. rather less than pr/(u—1)). It is easy Wide Angle Lens for Cloud Recording. 1001 to show that a cloud at an angle @ from the zenith will appear on the plate PP at a distance w from the centre of the image, given by w=hsin 0/ Vu?—sin? 0. The lens is, of course, not corrected for chromatic aber- ration, but this might be largely removed by employing a monochromatic filter, which could be closen so as to facilitate the photographing of clouds. The finite size of the image of a distant point source, due to the plate PP not coinciding with the true image ITT, ean be shown to be very approximately d/7, where d is the diameter of the aperture in the screen 8. If the hole in the screen be of diameter d=7r/20, the aperture of the lens is roughly f. 50; and the finite size of the image mentioned above results in a blurring at the zones which are most out of focus, which is equivalent to an uncertainty in 6 of about +°. The illumination of PP becomes less for large values of 0, but the effect does not sem sufficiently pronounced to be objectionable in cloud photography, and need not be considered in detail. The angle of the field of view of the lens could be increased yet further by replacing the outer plane plate G by a plano-convex lens, having its curved face outwards. This arrangement might be used if it were desired that the photograph should show the horizon and slightly below it. An advantage of this arrangement would be that the zone for which the image is most cramped together would not be at between 80° and 90° from the zenith, but say from 95° to 105°, thus enabling the record of clouds near the horizon to approach more nearly the clearness of that of clouds at the zenith than would be the case if a simple hemispherical lens were used. It may be mentioned that the lens gives views similar to those seen by a fish in water. The apparatus has, however, probably been reduced to the simplest form advisable, though a less perfect image could be obtained by placing the photo- graphic plate in contact with one side of a thick parallel plate of glass, the other side being covered by a thin screen pierced by a small hole. Finally, it will be seen that when using the simple hemi- spherical lens, or the thick plate of glass just described, only two constants (viz. h and u) need be known accurately, to enable a complete network of degrees to be constructed for use in interpreting the photographs. Plul. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 3T 1002" | XC. A Problem in Viscosity: The thickness of liquid jilms formed on solid surfaces under dynamic conditions. By The Research Staff of the General Electric Company Litd., London. (Work conducted by F. 8. GoUCcHER and ie Warpoy+ ; |Plate VIIL.] Summary. (ake problem of determining the thickness of the liquid layer coating a solid body drawn out of a liquid is discussed theoretically and practically. It is shown that if the solid is a flat slab of infinite width, the forces deter- mining the thickness are those of gravity (g) and viscosity (n), and that the relation between thickness ‘¢), density (p), 209 | py Tf the solid is a fine wire of radius 7, surface tension is dominant and gravity negligible. If y is the surface tension, the relation must be of the form : =/(®). It is found and velocity of drawing v% is ? = empirically that ee is of the form 4°8 in ¢.g.s. units. These results apply to suspensions if (1) the diameter of the suspended particles is not greater than ¢t, (2) the effect of the particles in increasing 7 is taken into account. There is no evidence of any special cohesion between solids and liquids wetted by them ocher than that which prevents slipping at the interface. In many important industrial processes solid surfaces are coated with a layer of liquid by drawing them out of a bath of the liquid. The enamelling of wires or tubes is such a process, and so in all essentials is painting with a brush. We are aware of no theory or even complete experimental investigation directed to determine how the thickness of the liquid layer produced in such circumstances varies with the properties of the liquid, the solid surface, the velocity of drawing, and other possible factors. The experiments described in this paper show that the matter is surprisingly simple. * Communicated by the Director. A Problem in Viscosity. 1003 I. Theory. The factors which must be effective appear at once from fig. 1, which shows a longitudinal section of the solid A drawn vertically with velocity v, out of the liquid B. Observation shows at once that at least in some circum- stances, the thickness of the liquid layer C is constant for a distance above B very much greater than t. Consequently it is reasonable to assume that in C the stream lines are vertical ; and if they are vertical, continuity requires that the velocity v along any one stream line is constant. This constant velocity must represent a balance Fig. 1. (63) > ROSSER RCN SSSA SANS SAN RM SUMO MMO between the forces acting on any element of the layer, and of these forces one must be that due to viscosity of the liquid and another that due to gravity. If the solid is a plain slab of infivite breadth perpendicular to the diagram, it is difficult to see what other force can act; it is possible that, if it is sufficiently small, some special force of cohesion between liquid and solid is effective, but we shall’ see that the facts can be explained adequately without introducing such a force. If, on the other hand, A is a cylinder, surface tension may be effective ; for, owing to the curvature of the ae D2 1004 Research Staff of the G. E. C., London, on outer surface of UC, this tension will produce an increase of pressure in C which will not be balanced by any corre- sponding pressure over the ends of the layer. Accordingly, the liquid in the layer will be squeezed out of it at the lower end, and, possibly, at the upper. The balance of these forces must be such that the outer layer of C is at rest relatively to the liquid B in order that the continuity of the liquid surface may be preserved. This last condition may appear puzzling; for if the solid ° is continually moving upwards carrying the liquid C with it, it would seem that the outer layer of this liquid must be moving upwards. What really happens is that the inner layer, next to the solid, moves upward with the velocity of the solid ; the other layers nove upwards with a velocity con- tinually decreasing outwards, the difference between the velocities of different layers providing viscous forces neces- sary to counteract gravity or surface tension. If the layer at distance « from the solid moves upward with velocity v, it will require a time //v after the drawing starts before a layer of thickness w is found at a height / above the liquid. Strictly speaking, it will require an infinite time before the layer of full thickness, corresponding to v= 0, appears at a finite height above the surface. “But a consideration of the numerical values in the equation about to be deduced will show that the time required fora layer of thickness differing | inappreciably from « to form at a distance of severai centi- metres above the surface amounts only to a few seconds. Accordingly, if we wait a few seconds between starting the drawing and taking observations the thickness of the liquid on the surface will be practically equal to that corresponding oO 0 With these considerations in mind the complete solution of the case of the infinite slab is easy. If x be taken horizontal and z vertical, with the origin on the surface of the slab ; and if p is the density, 7 the viscosity of the liquid, v its velocity relative to B, we have d dv | ae (1 <) py fey . A ho) (1) with the boundary condition that at «=t, - =O) a Hence v= a — tw) +05 52.) 7 ee) at w=0 if there is no slip v=vp, the velocity with which the ‘ a Problem in Viscosity. 1005 slab is drawn upwards; ¢ must adjust itself so that this condition is fulfilled, 7. e. ae tan/ 2 i LS ae) It is not easy to measure ¢ accurately while the slab is moving, and of course, if it is stopped, the conditions are changed at once. In practice the liquid layer usually sets, owing to cooling, evaporation, etc., at some little distance about the liquid surface. In our experiments we have used a liquid with a melting-point above room temperature, so that it freezes on the slab a little distance above the surface of the bath; we can then measure the thickness ¢’ of the solid film. If in these conditions the assumptions we have made so far are legitimate, we have from the equation of continuity t y= | ede, B pateseeupenn ey ec! «) CAs) 0 2v0n = FSU am AC) ee, Se eee (0D) doe (5) But the assumptions cannot be accurately true, for since the thickness of the layer decreases as the liquid cools and a3 its viscosity increases, the stream lines cannot be vertical or of constant velocity. We shall consider the effect of this failure of the assumption in III. If the surface were a circular cylinder of radius 7) we should have instead of (1) d dv a7.) = pg ME Risto). hee CO} which, using the same assumptions as before, gives in place of (2) ye — 2 ” vant 4! 9 : — (tr)? log k. Sea) t When za is small (7) gives which gives i Cole g : was e as in (3). > t . : . . If " is not small we may expect surface tension forces to Y I 0 be appreciable. The calculation is then more difficult and we can give no complete solution. The flow would appear to depend on the exact form of the meniscus at the surface of the liquid. Buta dimensional argument gives us some information. 1006 ~— Research Staff of the G. E. C., London on If both gravitational and surface tension forces are appre- ciable, we must have | bg yee ov ; flee r, Y ? ae) — C, e ° ° . (8) where 7, are lengths and © a no-dimensional magnitude characteristic of the system. We shall see later that con- ditions can be found in which the gravitational forces are inappreciable compared with those due to surface tension. If we assume that this condition is fulfilled and assume further that there is only one independent length 7,, viz. 7, then (8) reduces to ee hl G - J ah ( ) A ea rat Cat Sele! (s) It should be pointed out, however, that the last assumption is precarious. For when surface tension is effective, 2) the height of the meniscus over which the pressure due to surface tension varies from ¥(7)-t) to zero is likely to be as important as 79. (8) can be valid over the whole range only if 2 is proportional to 7. But if it is true, we should expect the importance of the surface tension relative to the gravitational terms to be measured by 7 so that the gravitational terms may be neglected when this expression is large. The dimensional argument cannot, of course, prove that the distribution of the liquid as an even layer is stable. The analogy of a water jet strongly suggests that it will not be stable. If it is not stable, the conditions may be entirely altered by freezing the layer as soon as it is formed. This possibility will be considered later; but if they are not greatly altered, the effect of freezing will merely be, as before, to introduce a constant factor. The form of (9) will not be changed. Another ease of some interest may be mentioned briefly, although we have not investigated it experimentally. If a eylindrical tube is drawn out of the liquid, the gravitational and surface tension forces will act in the same direction in the layer on the outside of the tube and will act in opposite directions in the layer on the inside. If the dimensions of the tube are such that the two sets of forces are comparable, the layer will therefore be thicker on the inside. It is interesting to speculate what will happen inside the tube when the surface tension forces are great compared with the gravitational. It is easy to see that in such conditions the . ae ee ae td < Fd a Problem in Viscosity. 1007 liquid must fill the entire cross-section of the tube, and will not begin to fall out of the tube till the column becomes so high that the gravitational forces become appreciable and the conditions supposed are violated. II. Apparatus. In order to test these theoretical considerations arrange- ments were made to draw metal strips or wires at known speeds out of a stable liquid, the viscosity of which could be Fig. 2. NS Ss LZ LIL LC varied by change of temperature and which would solidify at ordinary temperatures, so that the thickness of the layer could be measured. . The apparatus was designed so that the viscosity of liquid and the surface tension could be measured under the condi- tions prevailing when the wire or plate was drawn from it. 1008 Research Staff of the G. EH. C., London, on The essential features were the viscosimeter, the surface tension apparatus, and the drawing device. (a) Viscosimeter.—The viscosity was measured directly by means of the torsional force exerted on a flat disk by another disk parallel and near to it when both are immersed in the liquid and when the second disk is made to rotate at a fixed speed. If the geometrical arrangement is unaltered the torsional force will be proportional to the viscosity of the liquid and to the velocity. A is a metal vessel containing the liquid and surrounded by a thermostat B containing glycerine. Through the bottom of this vessel passes a spindle C to which a pulley is fixed driven by a small motor. To the top of © is attached the rotating disk E in the centre of which is a jewelled bearing. Above this is the fixed disk separated from it and supported by a hardened steel point resting in the jewelled bearing. A light steel spindle F passing loosely through the metal cover serves to keep the fixed disk parallel to E and to connect it with a torsion measuring device consisting of a spiral steel spring H and an aluminium pointer G which moves over a graduated scale on the metal cover. Vanes D are also attached to the spindle C which serve to stir the liquid by causing it to rise through vertical channels in the metal cylinder K and to fall over the top of K back into the bath. The bottom of K is bevelled as shown, so- that if the liquid contains suspensions these will fall in between the vanes and thus be kept from settling. _ The thermostat can be heated by means of the gas ring burner placed below it. (a) The viscosimeter was calibrated by means of solutions of known viscosity. A 60 per cent. sucrose solution was found exceedingly useful] for this purpose, as it gives a wide range of viscosities with temperature change, and the viscosities at various temperatures have been accurately determined by the Bureau of Standards *. It was found that the scale deflexions for a given speed of the rotating disk were directly proportional to the viscosity, and also that for a given viscosity the scale deflexions were directly proportional to the speed of the rotating disk overa wide range. (0) Surface Tension Apparatus——The surface tension apparatus was measured by Wilhelmy’s method, viz. by means of the force required to break the film drawn out of * It. C. Bingham and R. F. Jackson, Bureau of Standards Bulletin, “No, 14) p59 917): a Problem in Viscosity. 1009 the liquid by a given length of straight fine wire. If / is the length of the film, f the force required to break it, then y= a Fig. 3 is a perspective view of this apparatus, which was mounted above the viscosimeter. A platinum wire frame of the form shown served to pull the film from the liquid, the horns remaining beneath the surface of the liquid until the film broke. This was attached to one end of a light aluminium pointer supported in a metal frame by a torsion wire as shown. ‘This acted as a balance for measuring the force required to break the film. Fig. 3. Wire Frame \2) % a The metal frame could be rotated about an axis through A, thereby raising the platinum wire frame from the surface of the liquid until the film broke. The reading of the pointer could be noted at the moment of breaking and the equivalent force determined by means of a small scale-pan and weights attached to the pointer in place of the wire frame. The restoring force of the torsion wire could be supple- mented by means of a rider attached to the aluminium pointer, so that for a wide range of surface ‘tensions the reading at the break-point could be brought on the scale. 1010 Research Staff of the G. E. C., London, on (c) Drawing Device.—The fine wires (tungsten or constan- tan) used as cylinders were cleaned superficially by heating in a reducing atmosphere. They were then wound on a bobbin P (see fig. 2) above the surface of the liquid, and drawn thence round an idle pulley L below the surface of the liquid and finally round the winding bobbin N, which was rotated at a regulated speed. Asan approximation to an infinite slab a copper strip about 1 inch wide and 0°05 em. thick was used. It was thought that if such strip were drawn through the liquid from a bobbin outside it, the liquid might be cooled appreciably when the drawing was rapid. Accordingly the strip was originally wound on a bobbin, wholly immersed in the liquid, and drawn thence direct to the winding bobbin. 7 (d) Estimation of thickness—The thickness of the layer on the fine wires was determined by weighing a known length of the coated wires on a torsion balance (designed in these Laboratories) capable of estimating a few milligrams. with an error of 0:0001 milligram. The coating was then dissolved off and the bare wire weighed. The layer on the strip is uniform only at some distance from the edges ; accordingly a known area was cut from the central portion and weighed before and after removal of the layer. The density of the solid layer was determined on a sample of convenient size. : (e) Liquids used for coating.—Molten waxes were first used as the liquid for coating the solid surfaces. It was found that filtered beeswax and carnauba wax were most suitable. A range of viscosity from 1 to 100 centipoises was. obtainable within a temperature range from 50° to 140° C. In some experiments the viscosity was increased by adding fine insoluble powders to form suspensions. Waxes were expected to he particularly advantageous, because their surface tension is low and therefore not likely to be changed by grease and other impurities of low surface tension. But it was found in the course of the work that aqueous solutions could be used with convenience ; for by the process of drawing surface impurities were removed very rapidly, so that after a short length had been drawn the surface tension was con- stant and normal. Some of the observations recorded were made with aqueous solutions of sodium silicate, which were dried on the wire by passing it through a small electric furnace a short distance above the liquid. a Problem in Viscosity. 1011 III. Eeperimental Results. The results of the experiments on flat strips are shown in fio. 4, where ¢ {estimated from ¢' by (5)} is plotted against 3 7 ae from 0°122-0°320 c.g.s. units, v) from 2-46-7°85 cm. per sec., p from 0°8-1°2. ~The straight line is: calculated from (3). It was pointed out in I. that the assumptions used in the derivation of equation (5) could not be accurately true. Owing to the cooling of the wax, as soon as it leaves the liquid surface its viscosity is increasing, so that it is to be expected that the effective viscosity will be somewhat larger Fig. 4. Ee ee hs oly, MR DEUS SESE e IIR aes eee Cy BZe paleg pele ade ead pt ala Lb Tey ng a See ig at aad ia) 2 a a al SE ee eee | ae Le Vr than that measured. The observed thickness of film should therefore be thicker than that calculated on the basis of the measured value, as is actually found. But the difference is comparatively small; and when it is remembered that (3) involves no empirical constant, but is calculated from measured values only, no doubt will be felt that the theory offered is completely confirmed. The coating on the edges of the strip was much thinner than on the central portion ; ; it was therefore to be expected that surface-tension effects \ ould be prcminent in fine wires. Preliminary measurements on the thickness of the coating on wires of diameter less than 0°02 mm. showed that the coatings were much thinner than were demanded by the theory which takes only gravitation into account (eqn. (7)). 1012 Research Staff of the G. EH. C., London, on Moreover, it made no difference to the thickness of the coating whether the. wire was drawn vertically from the liquid or horizontally through the top of the surface formed by a rotating wheel within the liquid. Surface tension must therefore be the dominant force. This result is in in the accordance with eqn. (9), for the least value of _ ; observations was 300. PI However, the possible effect of instability due to surface tension must be considered. It was clear immediately that the layer originally formed was unstable, for the solidified coating on the wire was not even but was broken up into blobs. These blobs were beautifully symmetrical and evenly distributed when viewed under a microscope. Photomicro- graphs (1), (2), (3) are shown in fig. 5 (PI. VIII). But formation of the blobs evidently occurs after the wire has left the liquid surface ; for it does not depend on the velocity of drawing or the viscosity within fairly wide limits. No. (1) was obtained at one-half the speed of No. (2) and at twice the viscosity ; in these two cases there is approximately the same thickness of coating. No. 3 was obtained at a higher speed than No. 2, but at the same viscosity. Here the coating is obviously thicker. That the blobbing took place after the liquid coating had been withdrawn from the liquid was further demonstrated by drawing a wire through glycerine; a very thick film was obtained which remained smooth and even for some inches above the surface of the liquid before it could be seen to break up. A systematic study of the film thickness as a function of viscosity and surface tension and velocity of drawing as well as the radius of the wire was undertaken. The following were the limits of the variables :— Radius of wire r=0:00075-0-01 em. Velocity of drawing v=5°5—-66'0 cm./sec. Viscosity 7 =1-100 centipoises. Surface tension y= 36-62 dyne/em. Density p=0°8-1'4 grm /cm’. The limits of v and 7 that could be investigated were set, on the one hand by the impossibility of measuring very thin layers, on the other by the thickness of the film; if ¢ were greater than 79 the blobs which formed fell off the wire. ‘The results are shown in fig. 6 hy plotting ( ~) against (7) a Problem in Viscosity. 1013. If the theory given is correct, the points should lie on a smooth curve. They do actually lie on such a curve within experimental error, and the curve is seen to be very nearly a straight line. All our measurements on wires can simply be expr ressed by the formula eee ade py (10) the constant 4°8 above being empirical. It appears, then, that the two extreme cases considered theoretically, namely , all gravity and all surface-tension, are easily realized. Indeed, we have not realized an intermediate ease, although it would probably be possible to obtain ira SoeSee00ee06 460 AN! a it with cy ‘linders of greater radius. Further, it appears that there is no special force of ‘ cohesion’ between solid and liquid surfaces which can produce a film even as thick as the thinnest we have examined (0:00007 cm.). If v were- infinitely small, 7. e., if the surface were allowed to drain for an infinite tame: there is no evidence that the film would not be completely removed, or at least reduced to molecular thickness. The only action between solid and liquid im- portant in these experiments is that which prevents slip at the interface. It follows that the coating should be in- dependent of the surface of the solid so lone as the liquid wets it at all. As far as our experience goes ; the conclusion 1014 Prof. 8. Timoshenko on the Distribution of is correct ; e.g., a tungsten wire coated with graphite (from the process of wire-drawing) gave the same results as a -clean wire. In all the experiments described so far the liquids used have been true liquids. But in such processes as enamelling or painting the liquids are usually suspensions. The varia- tion of the viscosity of suspensions with their solid contents cand with the size of the suspended particles has been investigated by several authors*. We have repeated some of this work on liquids in which we were particularly interested, and have confirmed many of their results. But the question arises whether the viscosity of a suspension measured by shearing it between parallel plates is the same ‘as that which determines the amount of liquid adhering to a solid drawn out of it. We have made many observations in this matter. It appears, as might be expected, that the two viscosities are the same so long as the diameter of the suspended particles is not larger than the thickness cf the liquid layer drawn out. If the diameter exceeds that thickness the liquid behaves in drawing as if it had a viscosity much less than that measured by shearing. But a -consideration of fig. 1 shows that such large particles cannot be expected to enter the layer of liquid on the solid surface ; they are squeezed out from it. Accordingly © the failure of formula (10) for these large particles is simply due to the fact that the liquid which is being drawn is that from which the large particles have been removed -and of which the viscosity is correspondingly lower. XCI. On the Distribution of Stresses in a Circular Ring compressed by Two Forces acting along a Diameter. By S. TIMOSHENKO fF. Cee. the problem as a two-dimensional one, we ‘/ can obtain a solution in the case represented in fig. 1 ‘by combining the known solutions of the problem of com- pression of a disk f (fig. 2) and that of a ring § (fig. 3). If we take the normal and the tangential tensions acting -on the inner rim of the ring (fig. 3) as equal and opposite to the tensions acting on the cylindrical surface of the radius r in a disk (fig. 2), the stress-distribution in the case of * KE. C. Bingham, Bur. Stand. Bull. no. 278 (1916). %. Humphrey -and KX. Hatschek, Phys. Soc. Proc. xxviil. p. 274 (1916). + Communicated by Prof. HE. G. Coker, F. RS. . { See A. E. H. Love, ‘Treatise on the Theory of Elasticity,’ p. 215, 1920. § A. Timve, Z. f. Math. u. Phys. liu. p. 848 (1905). Stresses in a Circular Ring. 1015 fig. 1 will be obtained by summing the stresses corresponding to figs. 2 and 3. Fig. 1. Fig. 2. £ <—t— eS ee Fig. 4 shows the normal stresses on the vertical and horizontal cross-sections of a ring when R=2r calculated by the above method. The dotted lines on the same figure represent the results of elementary solutions obtained by using (1) the hypothesis of plane cross-sections or (2) the hypothesis of plane distribution of normal stresses. The stress-distribution in a compressed disk (fig. 2) is obtained by superposing a tension * pe = (Ae on) eee | * We suppose the thickness of the ring equal to unity, 1016 Prof, §. Timoshenko on the Distribution of equal at all points, and two simple radial distributions : __ 2P cos dy 2P cos ds i On Tepe #8 EF Ag S| A AADs| | | BRAC Ee NZ WL) Teese HE ee Bee If R=2r, the corresponding normal and tangential stresses at the points of the cylindrical surface of the PEE r ae be approximately represented by the following series * pe oH (0°506 + 1:008 cos 20 + 0°443 cos 40 + 0°158 cos 68 + 0°0467 cos 86 + 0:0083 cos 108), = ap (3) 10 = =e (0°749 sin 20+ 0°374 sin 40 + 0°141 sin 60 +0°0460 sin 80 + 0:0133 sin 108). Distributing on the inner rim of the ring (fig. 3) the tensions * The calculations have been made by the Runge approximate method. See Theorie und Praxis der Revhen, p. 158. rom the calculations made we may conclude that the error in stresses will not be higher than 4 per cent, if we take the first six terms of the series only. Stresses in a Circular Ring. 1017 which are equal and opposite to the tractions (3), and using the solution of the ease of a circular ring, the following expressions are obtained for the stresses at any point :— 2P As 7 R? + p? 90 eo R| — 0506, ae (22686-3246, aa oR, t R? 5, Rt 2 ies +0-4832— Jeos 20 +( 0°3691 a —0°6783 F ajeg RS RA p! +0 0368 — —0-0599 =, ) 0s 40 +(0-06504 on p p Rl 6 8 6 —0-10026 F. + 0-0041319 S —0-009524% RS eos 68 5 8 10 (4) +(0-008758-%, —_0-01225 8, = fp + 000080795, RS) ; ~0-0010888 =) ena (0-0007880 F, | | pr? sete pce, —0-001037-£-, + 000002960 = RI0 —0°00008475 =) cos 100 |, Peer Re Ree ae ee . Ra a SES ae —(*- : . rae [0 506", —a 268 — 04832 Ry -— 285" AAR ray tron root | +-2°752 = cos 20 +(—0-3691 = +0 22615, 4 R¢4 (5) 0 0 : 3U 6 8 6 +0:05013 Ff —0-0041319~ 4 0-01904**, p p , pe ie Ri + (—0-008758 pe + 0°007352 £ —0-00040795 a 5 4 —0-0368—% +0'1798— cos AO + (-0-06504 ) cos 60 R8 p® a. 00018146" 603 80. { —0-0007880 = a ae Sie RY +0-0006911 frp —0-00002960 R09 +0:0001265 = cos 108 |, Phil. Mag. 8. 6. Vol. 44. No. 263. Nov. 1922. 1018 Prof. 8. Timoshenko on the Distribution of oe zeal alc 268—3'162 &, —0-4839 e R? +1°376~; )sin 20+ (0: 36918, =i) — RS —0:03680 ° +0: 1198) sin AO + (0° 065046, Re —0-07520 F - —0-0041319~ e+ 001428 = sin 68 i (0 008758-2, —0 009802 BK: ele Ré hh. RS +0:0014517 #) sin 80+ (0 0007880 4, —0: 0008638 67, i) 00002960 + 0:0001054 a sin 106 | k where p denotes the distance from the centre. Superposing the stresses (4), (5), (6) on the stresses (1) and (2) above, we obtain the stress-distribution in the com- pressed ring of fig. 1. The normal stresses on the vertical and horizontal cross-sections of the ring have the following values :— me I? R? + p” o ie R? 8 p' —0°6133 5; -—0-0915 ? = _—0 01146 £,—0- 001045, ser 4233 +0: 02728, +- 0°003043 = + 0: 000 Ey (68)_-0=— , 5-0-5065 nes + 22685-95525 + oono200 go a a dee Ue ag ae aoe (R?+p2)? 0°506 — Pe R—P mm te: be —0°7433 € +0°1090 ie =i); 013048, R! RE Rs 6 OOO SM te fete OG eee 005221 —, R10 p° ite ae a 8 (60), — 2-268 Stressesin a Circular Ring. 1019 The results of these calculations are represented on fig. 4. As a measure of the accuracy of these calculations, we have R ( (40), xdp=—0-s022P, . be pert al.) ey ree so that the error is smaller than } per cent. If the corresponding calculations are made for the vertical cross-section, we obtain R P (0), do= 9p 01996, fe es laiiewera This must be compared with the result i 2 cos g@ sin ddd= yi ; 0 a corresponding to a’ simple radial distribution (fig. 5). We have again an error smaller than 4 per cent. The method outlined here may be applied to the more general case of a circular ring subjected to any forces acting on its external rim. It is oniy necessary to use the corre- sponding solution for a disk *. | * A. E. H. Love, ‘ Treatise on the Theory of [lasticity,’ p. 216. 3U 2 S(1020 ). XCIL. On a Revised Equation of State. By ALFRED } W, Porvur, 05Sc.. HRS: Ff nst.eo pee See eran equation of gases in the reduced form is 1 Gao | xP: | 2(1- ph which can be modified by putting y” instead of y in the exponential term. This equation is very fairly satisfactory, when n=3/2, in the region of low pressures; but it breaks down for pressures above the critical value. Berthelot also developed an .equation in which the respective terms are based directly on experiment instead of theory. It is explicitly applicable to low pressures only, and is very much used for that region. This equalion 1s 16 B2¢ | ay Bad It will be observed that it does not pass thr eee ‘hee critical point («=@S=y=1). Onfig, 1 are shown experimental values of «8 plotted against « for isopentane, and on this same figure values calculated from Berthelot’s equation are represented by a.dotted-curve. _ Values-from van der Waals’ equation are indicated by small squares. | The chief fault of Dieterici’s equation is that it makes the critical volume only twice the least volume of the liquid, whereas experiment shows that it is in most cases very nearly four times. To get over this, Dieterici treats the volume of the molecules themselves as being a function of the pressure. The first object of this paper is to point out that there is a way of testing the equation which shows that this last-stated modification (even if it should be necessary) cannot be the only change required, and that it is no use making it until other changes are made. : If the equation be written a=yl'(8) exp.(3). where F(@) is a function of the volume alone, the value of on * Communicated by the Author. Ona Revised Equation of State. 1021 y Ou Lb n eee fink “—~ —] becomes — —~ whatever the function (8) may be. a OY y"B ri ( ) Y Calling this quantity / (it is connected with the internal eile lg ie Cae REECE Che Rta ee id 2 iss | SN pe hey ie ol JS) oe ey enn A ey Ede Eee of 2. O © Experimental. + + Berthelot. C] () Vander Waals. —— Porter. latent heat of expansion of the gas per unit change of volume), the value of §/ should, according to the Dieterici 1022 Prof. A. W. Porter on a equation, be a constant. In fig. 2 are shown two curves of Gl plotted against the density (1/v) for isopentane de- termined from the experimental values. Curve A is the critical isotherm and B is the isotherm fer 503° abs. It will be seen from this figure that instead of being constant it undergoes very considerable variation. Near the critical point the calculated points fluctuate, but a smooth curve drawn amongst them shows that an equation of at /east the second degree is required to represent them. Fig. 2. Ae Es Ss) Geez ae iS ee ae a es —| Sel ees al en arene = =a ; Isopentane. (Oe ) ” plotted against Density. ( aT ie p P 8 y, Curve A, Critical [sothermal, T = 461 abs. C. », 3B, Isothermai, T = 503 abs. C. If the necessary extra terms are introduced, it becomes possible to bring the equation for the critical isotherm into the form The three conditions for the critical point are satisfied if N=33, G—iol. b="261,.. c= —"Zare It is noteworthy that the negative value of ¢ is required not only from the data of the critical point, but also by the isotherm on fig. 2. Revised Kquation of State. 1023 The values of a, 6, and ¢ will be functions of the tempera- ture. 1 O(a —l a DIT ROC At 2=0,8=«a, this becomes zero (and because a does not do so, so also does 0 (a@8)/d«) when Gis. If we write 0 a 4 n this occurs when y” =4a,= 6°04 = (2°455)?. Now the e#@ against « curve for nitrogen starts out horizontally when y=2°54; hence n=2 nearly. Inserting these values, which are obtained solely by making the equation suit the critical state, it is interestiny to see how nearly the equation becomes Berthelot’s equation when the pressure is small. It can be written vi . eee | (1-+ = aa Caw ye 5°28 ar aye. | 0 — ) = for large values of ~. Ce 5 bac ia i In Berthelot’s equation the numbers are 95°33 and 3°95 respectively. If 4 be written 6,/y”, the value of mis found from the curves of isopentane to be a high one—about 12 to 15. It can be obtained also by considering the value of ge at the eritical point. We have in general y Oa nay 4. mb, 6 “OY BR VG 4 MER, or at the critical point the right side is 1+na,+mb,+9¢, or 14+3+°267m—°237q. There are not data enough to find m and q with certainty. But since this critical slope is for all substances nearly equal to 7, it follows that m must be at least 10, which agrees with what we find from the experimental curves. 1024 Prof. V. Karapetoff on General Equations The critical isotherm calculated from the values of the constants obtained above is shown in fig. 1 as a continuous curve. How nearly it fits the measurements for isopentane down to about 4 the critical volume is seen by examining the circles which represent experimental points. The following values given by experiment and also by various equations for the case of 8 =4 are useful for comparison to show the success and defect of the revised equation. Experiment. | Van der Waals. —_—Diieterici. | Porter. eas ILS) 4 | Infinity | 156 a38=1:07 © 2 | “Infinity 9) ee XCIII. General Equations of a Balanced Aliernating-Current Bridge. By Vuapimirn Karapetorr, Professor of Electrical Engineering, Cornell University, Ithaca, N.Y.” N the last few years the use of the Wheatstone bridge for the measurement and comparison of inductances and capacities has cousiderably increased, partly due to developments in the art of electrical communication, and partly because of improvements in the sources of high- frequency sinusoidal currents. Old classical arrangements of alternating-current bridges have been further developed and new arrangements evolved t. This seems, therefore, to be an opportune time to deduce a general equation of the a.c. bridge which would comprise the various actual bridge arrangements as specific cases. Such a general formula gives a bird’s-eye view of the existing bridge connexions and will. enable new bridge arrangements to be devised without deducing fundamental equations in each case or constructing vector diagrams. It is hoped that the general formula (D) given below will serve these two purposes. Some time ago Dr. Poole showed { that the currents and voltages in the usual arrangements of an a.c. bridge, when balanced, can be represented by comparatively simple vector diagrams from which the relationship between the * Communicated by the Author. t See D. I. Cone, “ Bridge Methods for Alternating-Current Measure- ments,” Trans. Amer. Inst. El. Engrs. vol. xxxix. p. 1748 (1920). { H. H. Poole, “On the Use of Vector Methods in the Derivation of the Formule used in Inductance and Capacity Measurements,” Phil. Mag. vol. xl. p. 798 (1920). of a Balanced Alternating-Current Bridge. 1025 desired quantities can be readily deduced. While his results will be very valuable to the practical users of the bridge, the other side of the problem, that is, a generali- zation of the theory, may prove to be of interest to investigators of new possibilities of bridge connexions. Fig. 1 represents general connexions of an a.c. bridge, with an impedance in each branch, and a mutual inductance in each of the lower branches. The upper left-hand branch consists of two paths in parallel, and the galvanometer is connected at an intermediate point, A,’, of one of the paths. Z2> —— hee Golveanometer or Telepwone A.C. Source This is the arrangement used in the so-called Anderson bridge, and is included in the general scheme because of its further possibilities. The bridge is supposed to be balanced on alternating current, that is, the galvanometer current is supposed to be equal to zero. The current in the lower branches is denoted by Ij, that in the upper branches by I,.. Im the divided branch 2 the current through the lower path is denoted by I, so that the current through the other path is I,—I. The line current is I,+1, The impedances Z in the two left-hand branches are denoted with the subscripts 1 and 2, to agree with the sketches in Dr. Poole’s article. The right-hand quantities are provided with the subscripts 3 and 4, although Dr. Poole uses again the subscripts 1 and 2, except in his fig. 5, where the subscript 4 is introduced in the same place as in this article. 1026 Prof. V. Karapetoff on Greneral Hquations Assuming the currents and the impedances to be expressed as complex quantities, we have the following three funda- mental equations of the voltage drop in the parts of the | brid ge :— PAH e) XSi, 6 oe 1% leZg= liZe= Gy) Xa (GoYA=10 ee) Kliminating the currents from these equations, we obtain the following general relationship among the impedances of a balanced bridge :— (Z3— Xone) [(Ze aR Xin) 4 ve aXxmi( Le a ii) Ne (Z, ae [ (Zy+ Xms) (1+ Ya(Zo+Zs)) + YaLeZs5] . - (D) In this equation the admittance Y, is used in place of the impedance Za; the relationship between the two is Z,Y,=1. While in any special case eq. (D) may be applied directly, there are some typical special cases for which it is more convenient to write simplified formsof eq. (D). Hight such cases are considered below. (1) No mutual inductances and a single-path branch 2. This means that X»y=Xm3=0, and Y,=0; we then have a simple bridge consisting of four impedances. Hq. (D) becomes Lls=Lyly... 5 (2) No mutual inductance and a single-shunt branch 2 Here again X,,=X,3;=0 and Z.=0. The points A, and A,’ coincide, and Z, is in parallel with Z,. Hq. (D) becomes LL = LL, (1 + Y ily) ° . c ° (2) (3) No mutual inductance and a double shunt in branch 2 as shown in fig. 1. In this case the only simplification 1 is that X,,1= Xn3=0, and eq. (D) gives Ashig=Ly[Zi(1 + Ya(Z2+ Ze)) + YahAeZs].. - (3) (4) Mutual inductance in branch 3 only and no shunt in branch 2. We have X,,,;=0 and Y,=0; eq. (D) becomes (Z3—Xing)Zg=Z,(Zy+Xmg)- - + - (4) (5) Mutual inductance in branch 3 only and a single shunt in branch 2. In this ease X»4=Z,=0 and Z, is in parallel with Z,. Eq. (D) gives | . (Z;— Xing) Dy Ly [ (Zi, at Xs) al mm YaZe) | oinitye (5) of a Balanced Alternating- Current Bridge. 1027 (6) Mutual inductance in branch 1 only and no shunt in branch 2. We have X»3=0 and Y,=0; eq. (D) becomes Z3 (Ze 50 nl = (Z,— Xin) Za - eRe aie « (6) (7) Mutual inductance in branch 1 only and a single shunt in branch 2. In this case Xp3=Z,=0 and Za, is in -parallel with Z,. Hq. (D) gives Zs[ (Z_+ Xm) + YaXmZe] =(Z,—Xm)(1+ Yo%e)Z,. (7) (8) Mutual inductance in both branches, but the branch 2 is not shunted. In this case Y,=0, and eq. (D) becomes (Z3— Xmg)(Zo a Xm) a (Z, — Xm) (Ly + ona) ° (8) In the following table the special applications of the bridge are those discussed by Dr. Poole in the article mentioned above, and the references are to the figures in his article. All these applications are covered by the foregoing eight special eases of formula D, and the ease number is stated for each application. In some applications the four resistances of the bridge have first to be balanced with direct current, in other cases it is not feasible. This is indicated by “‘ yes” or “‘ no” in one of the columns. It will now be shown how readily the familiar formulee for the measurements shown in the table can be derived from the formule (1) to (8), all of which are specific cases of eq. (D). ‘The item numbers below refer to the items in the table, and in each case an equation is used as indicated in the table. All the results check with those given by Dr. Poole. Item No. 1: (73+ jols)ry=7;(",+joL,). Separating the real and the imaginary parts, gives pe ere (!.) eee ee ee | ED) These equations combined give ee ePalh a. a's) oe Like hie ee) Eq. (9) shows that the bridge must be balanced on direct current as well as on alternating current. SS] ce Prof. V. Karapetoff on General Equations 1028 tN) frpod- ey ery ol ey eQem/C— ry ol+ 4 Erol Fu ®rol+ oy “Q “Q0URTe | “ON “SLL JUBLING JOE S$, e[00g -tu09 aq OF SeTyIJURN’) ; ‘209, Aouenbo.a yy soul 3 “Ayrovdeo ‘|-10q809F | pue souvjonpul jenny | Aone) D (0 <2) Ayrovdeo pus soURJONPUL-Jpeg “Wosrepuy 9 | (9=94) £yoedvo | pure souvjonpul-jjeg | Uos1epuy 6) ‘soiqioedvo OMT | “Aqneg oq zs : "20URy -ONPUI-JfPS PUL TVNIN, *[[OMxvy]AL ‘2 "‘ge0UBINPUT [BUINU OMT, | “[[OMXBIAL G | ‘SOUBJONPUI-JJOS OMT, | “TOMXLT "Tl ‘parnsvent co peard *payeursito| “ony moma AG | ULa}] of a Balanced Alternating-Current Bridge. 1029 Item No. 2 (see Note at the end) : (73 +joL3;—)jo Ms) joM,=( +joL,—joM,)joMs;, from which L.M,=L,M; ° ° . . < . . (12) and r35M,=7,M, ; - : : . 5 ° : (13) These equations, combined, give | M,/M,=L,/L,=7,/7, a tete Lek” eet vie (14) The bridge cannot be balanced on direct current, but the ratio of the resistances must be equal to that of the self- inductances, before two mutual inductances can be compared. Tiem No. 3: (73 -+j@L3—joMs;) r=, (74 +-JoL,+joMs). Consequently 1309 =T3i"4 AEs i chs eed eras 9) ro(iz3 — Ms) =7,(Ly+M;3) ... . . (16) so that 1 /%2=1'3/74= (3 — M3)/Ly+M3). . . (17) Item No. 4: z : —jry/@C3= —77,/oCy oe Me ee vats ais tsp ais se wured ee |) CLS) tiem Nos 82 | (73 +jols)r2=7y"4(1+772@Ca), from which To Tae NE Ak a ene ye a CLO) = FOr I OM et A (20) Item No. 6. r,(7r3+joL3) =r,[7,(1 +joCa(rs +15) +jpoCar or |. Kquating the real and the imaginary parts, we get LoV3 SP 14 : : - - “ ° ~ ° - ; (21) % Lis = 11Cq [ ral + 1p) + Tots | a - : (22) or, combining the two, Tg=Cobriret re(ti 73) }- (28) When 7;=0, this expression becomes identical with eq. (20). Item No. 7: (7'3—j/@C3) joM,=(7, +joL,—joM,)7. LOGOS a rote Vs. Karapetoff on General Equations Separating the real and the imaginary parts, we get : MO 3 erty sc Ae ee and 73M, = (L,;— M,)74. ohne Sepa ate 5 (25) Eq. (25) may be also written in the form L,/M, — 1 + (1°3/7°4) e e . . e (25 a) Item No. 8. This frequency bridge was described by - Mr. Cone, zbid. p. 1749. Eq. (2) gives :— rsrg= 147) —3/(@C,) | 1 +joCore). . . = (26) Separating the real and the imaginary parts, we get 1309/7 = Vy Se Cyr|C;, ° ‘ 6 5 0 (27) 1/(@U) =o. 2) 2 eee, The last equation gives w?C C9979 = 1 SVS (29) from which the unknown frequency may be computed. The following special case is of practical interest. In eq. (27) put r=, and re=2r;; then C)=2C,, and eq. (29) becomes aC, =aCoro= Y paraen s ietes . (30) As is mentioned above, the general equation (D), or any of its particular forms, (1) to (8), may be used for the derivation of new forms of the bridge. ‘lake for example the simplest case, that of eq. (1). It may be written in the form (13-1 ps) a +22) = (v7, +941) G4 je). . eee Separating the real and the imaginary parts, we get 13% g— Uglg=M— Vey. . . . . (82) U3Po + P3lg=&\rg+ V4 5 5 < 5 e (33) We may put, if we so choose, ryr.=r,7,, that is, require the bridge to be first balanced with direct current. Then eq. (32) becomes a3%,.=%,%,, and these two conditions, together with eq. (33), may be used to investigate various possible bridge combinations with resistances and induc- tances. ‘Ihe condition r3ry=r)r, may be dropped and eqs. (32) and (33) used for an investigation of various bridge arrangements containing resistance, inductance, and capacity. One or two of the 2s may be put equal to zero, with a resulting simplification in the algebraic relationships. Ina similar manner, eqs. (2) to (8) may be resolved into their of a Balanced Alternating-Current Bridge. 1031 component parts and the possibilities of various bridge con- nexions and measurements analysed, using only elementary algebraic transformations. When one or more branches of a bridge contain parallel paths, quicker results may be obtained by using admittances in place of impedances. Let, for example, the branches 1 and 4 contain ohmic resistances only, let branch 3 contain an impedance r,+j@L;3, and let branch 2 consist of a capacitive susceptance jwC, in parallel with a resistance ry. We then have, according to eq. (1), 1 foes (73 +)@]3)/ ie + jC Jann ee eae) rs +j@L3=ryr/t2+jorynC,. . . .. (35) Equating the real and the imaginary parts, eqs. (19) and (20) are obtained. or Fig. 2. Az Gel/vanometer Vote.—Fig. 2 shows the diagram of connexions of Maxwell’s mutual inductance bridge (item 2 in the table). At first sight it does not seem possible that fig. 2, with its two separate circuits, could be a particular case of fig. 1 which has one circuit only. In order to explain the transformation of fig. 1 into fig. 2, intermediate diagrams of connexions are shown in figs. 3 and 4. In fig. 3, the impedances Z, and Z, are assumed to be very small, otherwise the connexions are the same as in fig. 1 for the case of a single path in branch 2. In other words, the points O, B, and A, are electrically close to each other. In fie. 4 the limiting assumption is made that Z,=Z,=0, and the three points are brought together. Under these conditions 1032 Equations of Balanced Alternating-Current Bridge. no current can flow from the primary into the secondary circuit by conduction, since the two circuits have only one point incommon. The secondary current is produced only by induction, and the two circuits may just as well be entirely separate. In this manner fig. 2 is obtained from fig. 4. Fig. 3. 2> A> 24 Galvanometer AL Galvanometer This also explains the reason for which Z, and Z, in the table are marked equal to zero. ‘Thus, the fundamental formula (D) is also applicable to bridges in which the secondary current is produced entirely by mutual induction, and the primary source of current has no metallic connexion with the bridge itself. FE goaa. ] XCIV. The Motion of Electrons in Argon and in Hydrogen. By J. 8. Townsenn, M.A., Ft.S., Wykeham Professor of Physics, Oxford, and V. A. Battuy, W.A., The Queen’s College, Oxford *. i, HE experiments on the motion of electrons in argon which we have already published show such re- markable differences between this gas and nitrogen or hydrogen, that we considered it desirable to make further experiments with argon which had been very completely purified, and to extend the determinations of the velocities over larger ranges of pressures and forces. For this purpose it was necessary to construct an apparatus suitable for measuring the velocity in the direction of the electric force, and also the velocity of agitation when the electrons move in a widely diverging stream after passing through a narrow slit in a metal sheet. In order to obtain accurate results it is necessary in all cases that the gas should be free from impurities which tend to form ions. With gases like argon, where the electron loses a very small proportion of its energy in colliding with a molecule, the gas should be free not only trom impurities that tend to form ions but also from gases like nitrogen and hydrogen, as the loss of energy of an electron in a collision with a molecule of one of these gases, although small, is large compared with the loss of energy in a collision with argon. Also the effect of such impurities in argon is accentuated by the fact that the probability of a collision between an electron and a molecule is much greater in the other gases than in argon. It was found that impurities get into the gases from the materials such as ebonite or elastic cement generally used in the construction of apparatus for measuring velocities, so that in nitrogen or hydrogen the results obtained after the gas had been in the apparatus for a few days were slightly different from those obtained immediately after the gas had been admitted. In the case of argon the effect of these impurities was noticeable after the gas. had been in the apparatus for one day. 2. In order to eliminate the impurities emanating from the apparatus, glass was used instead of ebonite to insulate and fix in position the various electrodes and guard-rings, and the connexions were made through glass capillary tubes instead of ebonite plugs. The capillary tubes were slightly * Communicated by the Authors. Phil. Mag. 8. 6. Vol. 44. No. 263, Nov. 1922. 3X i =r 1034 Prof. Townsend and Mr. Bailey on the tapered and ground to fit into metal sockets in the outer case of the instrument, the wax used for sealing being applied only on the outside of the joint. A great improvement was thus obtained, and two instruments of different dimensions were constructed, one with a slit 2 centimetres from the receiving electrodes suitable for measur ing velocities in gases like argon where the lateral diffusion of a stream of electrons is very wide, and the other similar to that which had been previously used with the slit 4 centimetres from the receiving electrodes. When tested with hydrogen no change was observed in the velocities after the gas had been in the apparatus for several days, and with pure argon the changes in two or three days were extremely small. In the instrument with the slit 2 centimetres from the receiving electrodes the guard-rings and the electrodes were fixed in the positions shown in fig. 1. The electrons are set free from the copper plate P by ultraviolet light admitted through a quartz plate sealed in the cover of the instrument, and the stream of electrons that passes through the gauze G and the slit S is received by the electrodes H,, Hy, and Es. These electrodes were mounted on two strips of plate glass fixed to the guard-ring Rj,, so that the upper surfaces of the electrodes were in the same plane with the upper surface of the ring. The ring R, was 7:8 centimetres internal diameter and 11°6 centimetres external diameter, and was at zero potential. The ring Ry, of the same size as R,, was insulated and fixed at a distance of one centimetre from R,. The slit S was 2 millimetres wide and 1°5 centimetres long in a sheet of silver foil stretched inside the brass ring A, and fixed at a distance of 2 centimetres from the receiving electrodes. The gauze of silver wire G was at a distance of 3 centimetres and the plate P at a distance of 6 centimetres from the receiving electrodes. A uniform electric field was obtained by maintaining the ring Ry, the plate A, and the gauze G at potentials V, 2V, and 3V proportional to their distances from the receiving electrodes EH. In most of the experiments the plate P was maintained at the potential 6V, and the electric force from this plate to the gauze was the same as the force in the lower part of the field. The object of the gauze was to ensure that the electrons should have attained the steady state of motion corresponding to the force Z in the lower part of the field before passing through the slit. This condition may be obtained without the gauze by fixing the plate P at the potential 6V, and for experiments with gases at low pressures this gauze is unnecessary. But with large pressures above 20 or 30 millimetres the currents become Motion of Electrons in Argon and in Aydrogen. 1035 very small when the electric force is small and the plate P is at the potential 6V. The current is increased by increasing the pontential of the plate P, and with the gauze at the potential 3V the electrons pass through a distance of one centimetre under the force Z before reathing the slit. With the gases at the higher pressures the number of collisions of each electron with molecules of the gas in this distance is very large, and the motion of the electron acquires the steady state corresponding to the force Z before passing through the slit. Fig. 1 E A A SEES < 2) EY Le (ey ee R, E, See ae R, In the instrument with the slit 4 centimetres from the receiving electrodes there are three guard-rings between Ry and the ring A with the slit, as shown in the diagram, page 875, Phil. Mag. Dec. 1921. The dimensions of the electrodes and the guard-rings were the same in both instruments. In order to avoid errors which might arise from contact potentials at the surfaces of the rings R or the electrodes E, the metal surfaces were all electroplated with silver which is less liable to become oxidized than brass. The variations in contact potential which may arise owing to oxidation aX 2 1036 Prof. Townsend and Mr. Bailey on the would be too small to have an appreciable effect in most of the experiments as the electric forces were so large, but it was considered advisable to reduce as far as. possible an error that might affect the experiments with the smaller forces. : _ In the original apparatus the electrodes were of unsilvered brass, and when the experiments with hydrogen were repeated with the silvered electrodes almost exactly the same results were obtained. 3. The position of the slit in both instruments was adjusted to bring the centre of the stream slightly to one side of the centre of the electrode E,. This arrangement makes the instruments very adaptable for the measurement of the velocities in the direction of the electric force. For this purpose the stream is deflected by a transverse magnetic force H which is adjusted to bring the centre of the stream to coincide with one of the gaps between H, and the electrodes EK, and E;. The electrode EH, was a flat strip 4-5 millimetres wide and each of the: gaps ‘5 millimetre wide, so that the distance between the centres of the two gaps was 5 millimetres. Let this distance be 2a, 6b the distance of the centre of the stream from the centre of the electrode H, when H=0, H, the magnetic force required to deflect the centre of the stream through the distance a+), so that the current received by EH, is equal to that received by E, and E;, H; the magneticforce in the opposite direction which deflects the centre through the distance (a—b), the current received by Hi; being then equal to that received by KE, and H,. The velocity W in the direction of the electric force Z is given by the equations . H,W _ ath 7, ae C 5) e e e « e e Gi lel Wie | 7 Foo where ¢ is the distance from the slit to the electrodes EK. Thus 6 is determined by the relation H,/H;=(¢+6)/(a—0). The magnetic field which was uniform in the space between the slit and the electrodes EK was produced by a current in two large coils fixed in position on either side of the apparatus. With the larger gas pressures the velocities W are comparatively small, and it was convenient to deflect the stream through the shorter distance a—b as the coils became overheated when currents of the order of 15 amperes Motion of Electrons in Argon and in Hydrogen. 1037 were flowing through them during the time required to make the observations. The distance b was found with the gas at one of the lower pressures when the velocities were large, and comparatively small currents were required to obtain either the deflexion (a+b) or (a—b). This method was found quite satisfactory with nitrogen or hydrogen in the second instrument where the slit was 4 centimetres from the receiving electrodes, and the distance 6=°6 milli- metre. 4. With the first instrument where the slit was 2 centi- metres from the receiving electrodes, it would have been necessary to double the magnetic forces in order to produce similar deflexions. In this case the following method was used to measure the velocities in argon at the higher pressures. When the centre of the stream is at a distance b to the right of the electrode H, (figs. 1 & 2), the current received by H; is larger than that received by H,. By means of a suitable magnetic force H, the centre of the stream may be deflected through the distance 6 and thus brought to the centre of Hg. The two electrodes F K, and EB, then receive equal charges. The value of H, was found oF measuring these two charges with the central electrode maintained at zero potential, and adjusting the magnetic force to the point at which the charges are equal. The velocity W is then given by the equation Se ae ca cca tee C In order to find 6, the stream is deflected in the opposite direction through the distance (a—b), which is attained when the current received by E, and E, is equal to that received by H3, the required magnetic force H; being ee by equation (2). Thus b is given by the relation H,/H;=6/(a—b), and was found to be *87 millimetre. In argon at the higher pressures the velocities W were found by this method, ‘and the currents necessary to produce the magnetic fields H, were from 10 to 15 amperes. These results were tested by finding the velocities of electrons in hydrogen with both instruments. The hydrogen was admitted through palladium tubes sealed in the apparatus, and the experiments were made with different forces Z and pressures p, the ratio of the force to the pressure being varied from the value Z/p='2 to Z/p=40. There was a close agreement between the results obtained with the two instruments. The velocities obtained for the different values of the ratio en 1038 - Prof. Townsend and Mr. Bailey on the ’ Z/p were on an average less by about 2 per cent. than those obtained in the previous experiments * with hydrogen. _ 5. The velocity of agitation u of the electrons is deduced from measurements of the ratio of the charge received by the central electrode H, to the sum of the charges received by the electrodes H,, Hy, and Ez. The theory of the method has already been explained fF in detail, and may be expressed briefly as follows :—When a stream of electrons moves under an electric force, the number per cubie centimetre at any point is a function of the quantity eZ/mu* when the steady state corresponding to the force Z is attained, e being the atomic charge, Z the electric force and mu?/2 the energy of agitation of the electrons. If MO*[2 be the energy of agitation of a mole- cule of a gas at 15°C., the velocity of agitation wu of an electron would be 1: 15X10’ cm. per sec. if its energy of agitation were equal to MQ2/2. When moving under an electric force the energy of agitation of the electron is much ereater than this quantity, and if mu2=kMQ? the quantity eZ/mu? becomes eZ/kMQ?. This ratio may be written NeZ/kNMQ?, where N is the number of molecules per cubic centimetre of a gas at 760 mm. pressure and 15°C. ; and since the quantities Ne and NM©? are known accurately, the number of electrons per cubic centimetre at any point of the stream is a function of Z/k and known constants. The ratio R=n./(ny+ng+n3) of the charge nz received by the central electrode H, to the sum of the charges 1, ng, nz received by the three electrodes E,, E,, EH; may therefore be expressed in terms of the ratio Z/k, and the values of R corresponding to definite values of Z/k may be computed. The value of R for any value of Z/k may be represented by _ means of a curve, the form of the curve depending on the sizes of the receiving electrodes and the size and Pe of the slit. It was necessary therefore to calculate the values of R for different values of Z/k when the centre of the stream fell to one side of the centre of the electrode HK, with the slit two and four centimetres from the electrodes EK. As it is difficult to construct the apparatus so that the displace- ment 6 of the centre of the stream from the centre of K, is some exact fraction of a millimetre, the points on four curves were calculated which are given in fig. 3. The curves 1 give R for receiving electrodes of the dimen- sions shown in figs. 1 and 2, with the slit two centimetres * Phil. Mag. Dec. 1921. + J. S. Townsend, Proc. Roy. Soc. A, lxxxi. p. 464 (1908). Motion of Electrons in Argon and in Hydrogen. 1039 from the plane of the electrodes, the upper curve corre- “sponding to the case where the centre of the stream falls on the centre of H, (6=0) and the lower curve where the centre of the stream is one millimetre from the centre of H,(6='1). The curves 2 give the value of R under the same conditions, except that the slit is four centimetres from the receiving electrodes. In each case the curves for b=0 and b="1 are close together, and the correct ratio Rh for any intermediate value of } is easily estimated. Fig. 3, G-2 O-4 60:6 ory, 1-0 /-2 [4 1-6 The method adopted for calculating the ratios R corre- sponding to definite values of Z/k was similar to that used by Mackie * to find the points on the curve 2 (b=0). In the course of the calculations we redetermined the points on this curve and obtained numbers almost exactly the same as those given by Mackie. ~The calculated values of R, from which the curves were drawn, are given in Table I. TABLE I. : m | R. Ze k c=2 | ca—4 | 6=0 | b='l. b6=0 b= 05 196 195 Ven | he 1 231 230 ‘1607 1602 | 2 298 295 | ‘2077 2066 3 353 348 | 2495 2476 4 397 389 te, 5 435 “425 | 314 “310 1 Ser abt: “425 “415 It. 15 ie ald 506 -490 ae #8 aN | 570 548 | 25 eet ay ‘619 590 * J. H. Mackie, Proc. Roy. Soc. A, xc. p. 69 (1914). 1040 Prof. Townsend and Mr. Bailey on the Jf the ratio R=n,'Gy + 12+ nz) be determined experi- mentally when the stream is moving under an electric Force Z, the value of-Z/k corresponding to R is given by the curves (fig. 3), and the factor & is thus found. The velocity of agitation wu of the electrons is then given by the formuia u=1:15x10’* kk. None of the experiments were made with the ratio R less than *24, as greater accuracy is obtained with the larger ratios. 6. The accuracy of the normal distribution curves was tested by measuring the velocities of agitation of the electrons in hydrogen with each instrument. ‘The experi- ments were made over tke same range of forces and pressures as the test experiments on the velocities in the direction of the electric force. The values of & obtained with the two instruments were in very close agreement, and on an average they did not differ by more than 2 per cent. from the values of & found in the previous experiments. It may be mentioned that in the previous experiments the hydrogen was prepared by the electrolysis of barium hydrate, and passed over hot copper into a drying-flask, from which it was admitted through a tap into the apparatus, In the test experiments with the new instruments the hydrogen was admitted through a palladium tube without bringing the gas into contact with any chemicals from which an impurity might have been given off. There was no leak in either instrument which could be detected by means of a McLeod gauge, even when the apparatus was exhausted to 1/100th of a millimetre, and observations of the pressure were made at intervals during a fortnight. The results obtained with hydrogen may therefore be taken as being well established. 7. The argon used in these experiments was obtained from a cylinder supplied by the British Oxygen Company. The gas contained about 10 per cent. of nitrogen, which was removed by Rayleigh’s method. It was admitted to a vessel . containing a solution of caustic potash, and oxygen added in excess of the amount required to combine with the nitrogen. Two platinum electrodes were sealed into tubes leading into the vessel, and a discharge was passed between the electrodes for several hours. ‘The change of pressure in the gas due to the combination of the oxygen and nitrogen was noted, and after sparking for about fifteen hours the pressure was found to remain constant. The residual traces of nitrogen were removed by continuing the sparking for several hours. The gas was then passed slowly over hot copper-foil and into a drying vessel containing phosphorus pentoxide. Motion of Electrons in Argon and in Hydrogen. 1041 Two quantities of argon were thus prepared, one having had the traces of nitrogen removed by sparking for 120 hours and the other for 70 hours. These specimens of argon will be referred to as the first and second respectively. The velocities « and W were determined with both specimens over large ranges of electric forces and pressures. With the smaller pressures from 2 to 30 millimetres, where Z/p is large there was not much difference between the two specimens, but with the larg¢r pressures from 30 to 150 millimetres, where the range of the ratio Z/p was trom ‘1 to °8, there was a considerable difference. With these values of the ratio Z/p the velocities of agitation were greater, and the velocities in the direction of the electric force were smaller, in the first specimen than in the second. These results indicate the presence of a small trace of impurity in the second specimen. From our previous experiments we found that the loss of energy of an electron in a collision with a molecule of argon is much less than in a collision with a molecule of nitrogen or of any other impurity that the gas might be likely to contain. Small pee. of impurities have therefore the effect of reducing the velocity of agitation of the electrons, with the result that the velocities in the direction of the electric force are increased. 8. The following table gives examples of the experiments made with the first specimen of argon. The pressures p of the gas are given in millimetres of mercury, the electric force Z in volts per centimetre, and the velocity in the direction of the electric foree W in ecm. per sec. The quantity k is the factor by which the energy of agitation of an electron exceeds the energy of a molecule of a gas = a tO The velocities W and the factors & for electrons moving in argon and in hydrogen may be compared by the curves in figs. 4, 5, and 6. The curves (fig. 4) give the velocities W in argon and in hydrogen corresponding to the lower values of the ratio Z/p from ‘l to 2. The velocities corresponding to the larger values of Z/p are given in fig. 5 for argon, hydrogen, and a mixture of hydrogen and argon in the proportion by pressure of one of hy drogen to 24 of argon. In the ratio Z/p for the mixture, p is the partial pressure of the hydrogen. The values of & are given in fig. 6. There are two curves for each gas, the lower curves a giving k& for the smaller values of Z/p from °06 to 1:6, as indicated by the scale at the foot of the diagram, and the upper curves II for the larger 1042 Prof. Townsend and Mr. Bailey on the Tasue II. a Z. Zp. i. Wx10- 150 16° 0-112 96 100 10°5 0-105 95 “5 00 aes? “6168 | ae M0) 386 | oees, | ono 80 16:8 0-21 126 ae ae Aer ee ee 50 21 0-42 172 ee pe ae ee ee ey a ee EE oe 9:19 340) | is wee 63... | 920 | aoa Te Ol aoe aos OS ee OO Ge OT Gos 5g eh es sce) = a oe ee ee a On eee Coils em me = eae 125) locum eee Motion of Electrons in Argon and in Hydrogen. 1043: Fig. 6. ‘Z/p scale forcurves If 2 & é is /0 12 /4& 16 Z/p scale for curves I ——— 1044 — Prof. Townsend and Mr. Bailey on the values of Z/p up to 16, as indicated by the scale at the top of the diagram. Taking p as unity, the curves for argon show that as the force increases, £ increases rapidly and attains the value 340 when Z is 1°6 volts per centimetre, and after a diminution to 310 at 5 volts per centimetre, & rises again to 325 at 9 volts per centimetre and remains constant at — that value for the larger forces. J. The mean free path | of an electron may be obtained from the formula for the velocity W : | Wie coals oo ws a or mu This formula for the velocity of the electrons is obtained from Langevin’s more general formula for ions or electrons when the velocities of agitation are distributed about the mean velocity u according to Maxwell’s law, wu being the square root of the mean square of the velocities of agitation. Jt is difficult to determine the distribution in the case of electrons moving under an electric force, and according to Pidduck’s * calculations the factor *92 is more correct than 815, but the exact value of the numerical factor is uncertain, as the meanfree path depends on the velocity of the electron. The general conclusions obtained from the experiments as to the relative lengths of the free paths in different gases or the variations of the free paths with the velocity do not depend on the value attributed to the numerical factor in the formula, and as the value °815 has already been used in previous calculations, itis desirable to retain it for purposes of comparison. The effect of a collision on the velocity of an electron may be shown by calculating the coefficient of elasticity f by Pidduck’s formula. This method was adopted in the earlier researches on the motion of electrons in air +, and in those on oxygen, hydrogen, and nitrogen which were published recently {. . It is simpler, however, to give the proportion of the energy of an electron which is lost in a collision, as this quantity is found directly from the experimental results. The loss of energy of an electron in a collision may be estimated approximately from elementary. considerations. * ¥. B. Pidduck, Proc, ond: Math. Soc. vol. xv. pp, 87-12 (1915-16). + J. S. Townsend and A. T. Tizard, Proc. Roy. Soc. A, Ixxxvili. p. 336 (1913). t Phil. Mag, Dec. 1921. Pw Motion of Electrons in Argon and in Hydrogen. 1045 When moving along its free paths between collisions the mean velocity of an electron in the direction of the electric force is W ; and since all directions of motion are equally probable after a collision, the mean velocity in the direction of the force is zero after a collision and 2W before a collision. The loss of energy in a collision is therefore 2mW’. When variations in the mean free paths and the velocities are taken into consideration, it is found that the fraction A of its mean energy of agitation mu?/2 which an electron loses in a collision is given approximately by the formula V 72 A= 2°46 Paes e . . . . e (5) The following table gives the mean velocity of agitation w, and the velocity in the direction of the electric force in argon for different values of the ratio Z/p and the values of {and 2 obtained from the above formule. Since / and x depend directly on the energy of agitation which is pro- portional to k, the. values of Z/p are chosen to correspond to definite values of &. The values of J are for the gas at one millimetre pressure. | TABLE III. | { hiss 5s | Z/p. | WxX10-5. | ux10-7. ix10, | Ax 108 | 190 | a sa ee 115 20 1-79 | 120 | -195 3:9 12°6 14-7 164 | | 140 | ‘75 | ° 34 13°6 118 154. «| | 160 355 | 3:6 145 10°3 152 | | 180 | 440 3°85 15-4 9-44. 154 | | 200 525 4-15 163 9-02 Tee | 240 | 71 4°85 17°8 8-52 ea, | 980 | -95 6-0 19:3 Boe = See | 820 | 1:25 rar 20°6 888 | 3:45 mers 40 90-2 113 9:7 | 394 110 65 20°7 9-42 24-3 Piste 115 82 20°7 7:92 386 | i 10. The large values of k obtained in argon are due to the fact that the loss of energy of an electron in a collision with a molecule is extremely small, as shown by the figures in the last column. This loss is* very much less than in hydrogen or nitrogen. When moving with a velocity of agitation 12°6 x 10’ cm. per sec., the fraction of its energ lost by an electron in a collision with a molecule is 1'6 x 10~° in argon, 5x 10-? in nitrogen, and 4x 107? in hydrogen. 1046 Prof. Townsend and Mr. Bailey on the The increase of A with the electric foree when the mean velocity of agitation remains approximately constant at about 20 x 10’ cm. per sec. is clearly due to a large loss of energy in collisions with velocities greater than the mean, and a change in the distribution about the mean as Z and W increase. As an illustration of what would take place under this condition, it may be supposed that when the velocity of an electron exceeds a value A, its velocity is reduced to B when it collides with a molecule, and while its velocity of agitation is again increased from B to A, under the action of the electric force, the electron makes several] collisions with molecules in which there is very little loss of energy. The distance z that the electron travels in the direction of the electric force Z while the velocity of agitation rises from B to A is z=m(A?—B?)/2eZ, and the total number N of collisions with molecules while travelling the distance =z is approximately uz/IW. Hence N is inversely proportional to the product ZW. Hach collision in which there is a large loss of energy m(A?—B?)/2 is therefore followed by a ‘Jarge number N in which the loss is negligible, so that the average loss 1s inversely proportional to N and therefore directly proportional to ZW. Thus, although the inean velocity of agitation remains constant, the mean loss of energy in a collision increases with ZW. In this case the velocities of agitation are distributed near the mean value wu when Z and W are small, but as Z and W increase, the number of electrons with velocities near the mean diminishes and the number near the limits increases. | Another example of the effect of a change of distribution of the velocities of agitation about the mean, occurs when electrons move in pure hydrogen and in a mixture of argon and hydrogen. In hydrogen the loss of energy per collision is much greater for the larger velocities of agitation than for the smaller. Thus an effect which increases the number of electrons with velocities near the mean will reduce the average loss of energy per collision. With a constant force Z the velocity W in pure hydrogen is in many cases reduced by about 20 per cent. by adding argon to the hydrogen, while the mean velocity w of agitation remains unchanged. The loss of energy in the collisions with the argon may be neglected, so that in these cases the average loss of energy in collisions with molecules of hydrogen is proportional to ZW when the electrons are moving in pure hydrogen, and to ZW x°8 when the electrons are moving in the mixture, the reduction being due to a change in the Motion of Electrons in Argon and in Hydrogen. 1047 distribution of the velocities of agitation about the mean. In pure argon the velocity corresponding to k=340 may be taken asa lower limit to the velocity at which a large loss of energy occurs in a collision. ‘This velocity is the velocity due to a potential fall of 12°6 volts, and is a lower limit to the ionization potential. The increase in Xx 10° from 1°54 to 1:79 in pure argon when ux 107? changes from 13°6 to 11°5 may be due toa small quantity of impurity remaining in the gas. It will be noticed that the mean free path / changes from *118 cm. to *20 em. with this change in wu, so that the effect of an impurity would increase as wu diminishes, since the propertion of the total number of collisions in which there is a considerable loss of energy increases. 11. The mean free paths of the electrons are much longer in argon than in nitrogen or hydrogen. When moving with a velocity of 12°6 x “107 em. per sec., the values of / are *147 cm. in argon, ‘029 cm. in nitrogen, and -035 cm. in hydrogen, the gases being at one millimetre pressure. If the molecules were elastic spheres of the radius ¢ which is obtained from the viscosity of argon, the mean free path of the electron in argon at a millimetre pressure would be *0286 cm. With the range of velocities of agitation given in the table, the free path / in argon increases rapidly as the velocity diminishes, and much longer free paths would evidently be obtained if experiments were made with higher pressures and smaller forces. With the amount of pure argon at our disposal we were unable to make reliable experiments with values of Z/p less than °105, which gave k=95. The free paths given in the tables for the velocities 11°5 x10’ cm. per sec. and 12°6x 10’ cm. per sec. are probably too large, as may be seen by considering the effect of a large increase of / for a comparatively small reduction in w, on ‘the relation connecting W with Z,u, and Jl. If the Zel formula W= x 0°815 be taken as giving accurate values of / corresponding to the mean velocity of agitation « when a large change in wu produces a small change in J, the numerical factor must be increased when a small reduction in u produces a large increase in 7. The correction depends on the distribution of the velocities of agitation about the mean velocity u, and the rate of change of the mean free path with the velocity. When these two factors are taken into consideration, it 1s found that in the case of argon, 1048 Prof. Townsend and Mr. Bailey on the where the velocities u are about 12x10’ cm. per sec., the mean free paths obtained by the above formula may be 20 or 30 per cent. above their correct values, More accurate determinations of .the mean free paths in argon for these velocities of agitation may be deduced from the mean free paths in a mixture of argon and hydrogen; and it is of interest to compare the values of / obtained by the two methods. | 12. The simplest method of finding the mean free paths in argon when the velocity of agitation is less than 11'5 x 10" cm. per second, is to find the mean free paths in a mixture of hydrogen and argon and also in pure hydrogen, and to calculate thé mean free paths in pure argon from the two sets of measurements. The velocity of agitation is controlled mainly by the hydrogen ; and as there is so little loss of energy in the collisions with molecules of argon, the principal direct effect of the argon is to reduce the mean free paths of the electrons, and therefore to reduce the velocity in the direction of the electric force. In order to produce any measurable effect on the velocities of the electrons in hydrogen, it is necessary to add a large quantity of argon to it. In some previous experiments” it was found that when the partial pressure of the argon is four times that of the hydrogen, the velocities in the mixture were not more than 10 per cent. lower than the velocities under the same forces in the hydrogen alone. These observations show directly that the mean free path in argon for certain velocities of agitation of the electron must be of the order of fifty times the mean free paths in hydrogen at the same pressure. As no accurate conclusions could be deduced from experiments where the velccities differed by only a few per cent., the experiments were repeated, using much larger quantities of argon. The velocities W in the direction of the electric force for a mixture containing argon at a partial pressure twenty-four times that of the hydrogen are given in fig. 5, the values of Z/p being the ratio of the electric force to the partial pressure p of the hydrogen. Thus, taking p=1, the curves show that with a force of two volts per centimetre the velocity of the electrons in pure hydrogen at a millimetre pressure is reduced from 16 x 10° to 11-7 x 10° cm. per second by adding argon to bring the total pressure up to 25 milli- metres, The mean velocity of agitation is only reduced by 1 or 2 per cent. by the argon, so that under these conditions * Phil, Mag. June 1922, Motion of Electrons in Argon and in Hydrogen. 1049 the number of collisions of an electron with molecules of argon at 24 millimetres pressure is less than the number with molecules of hydrogen at one millimetre pressure. At the-higher forces the difference between the two gases is less marked. | The effect of adding 4 per cent. of hydrogen to pure argon may also be seen from the curves. Taking the case where Z is 16 and the pressure of argon 24 millimetres, the velocity W in pure argon is 46X10? cm. per second (as shown by the point on the curve for argon corresponding to Z/p="666). When hydrogen at one millimetre pressure is added, the velocity is 21 x 10° em. per second (corresponding to Z/p=16 on the curve for the mixture). The addition of the hydrogen causes the velocity of agitation to be reduced from 17°4X 107 to 8-22 x 107 em. per second, and this change in w would not be sufficient to account for the inerease in W if the mean free path in the argon were unaltered by the change in the velocity of agitation. 13. The following table gives the mean velocity of agitation u, and the velocity W in the direction of the electric force in a mixture containing 96 per cent. of argon and 4 per cent. of hydrogen by pressure, for different values of the ratio Z/p, Z being the electric force in volts per centimetre and p the partial pressure of the hydrogen. TABLE LV. | | & | *Z/p: |Wx10-%., wxl0-7. | Imx10% | &x10% | JgXx 102. _ 2s Ae ee ic) oa aie | | a 6:35 2:3 2°05 2 ea | ee he 75 7-95 2:82 2:09 339 138 mee tO Y:] 3:26 2°08 2:06 154 fama Te 1:28 10-0 3°64 1°99 3°88 161 | 13 1-72 111 4-15 1:87 2:66 154 16 2:95 12:2 46 1-75 2-49 138 20 3:25 13-6 B15 151 2-29 110 | 30 | 655 16-7 6:3 112 1:99 61:5 | 40 | 108 | 19-0 7-28 ‘89 1:95 | 39°5 Peels} 156; =| 20-9 8:14 76 200 | 29°5 Beg 260, |.) 23° 9-62 ‘61 2-30 20:0 1100 | 424 | 25:4 115 -48 2-98 136 140 | 64:8 | 26°5 13°6 39 4-00 10-4 The mean free paths corresponding to the velocity wu given in the last three columns of the table are: l,, for a mixture containing hydrogen at one millimetre pressure and argon at Phil. Mag. 8. 6. Vol. 44. No: 263. Nov. 1922. 3 Y 1050 Prof. Townsend and Mr. Bailey on the 24 millimetres pressure, / for pure hydrogen at one millimetre pressure, and J, for pure argon at one millimetre pressure. The free path J, is obtained from [,, and U, by means of the formula: oe aa The free paths J, and J,, are shown by the curves in fig. 7. Vig. 7. wma. meee a Bae’ dee lia: 14. The free paths in pure argon at a millimetre pressure are shown by the curves (a) and (6) (fig. 8). Curve’(a) for the lower velocities of agitation u, gives the free paths obtained from the measurements of velocities of electrons in a mixture of hydrogen and argon, and curve (0) the free paths obtained from the velocities in pure argon. The free paths tor the range of velocities from 11°5x 10" to 13-5 x 107 were found by both methods, and the curves tend to coincide with the larger velocities. Exact concordance can be expected only at points where the variation in the free path with the velocity is small, or where the velocities of agitation are very near the mean velocity w. For velocities between 4107 and 14x10’ there is a large increase in the free path as uw diminishes; and as explained in section 11, the free path calculated by formula (4) from Motion of Electrons in Argon and in Hydrogen. 1051 measurements of W and wv in pure argon is larger than the true value corresponding to the mean velocity w. The free path of an electron in argon at a millimetre pressure has a maximum value of 1°6 cm. for the velocity 3°75 x 107 em. per second, which corresponds to a potential fall of :39 volt. As the value of J is an average for a number of different velocities having a mean value w, it 1s probable that the mean free path for electrons all moving with the velocity 3°75x107 cm. per sec. is greater than 1°6 centimetres. . The free paths in hydrogen at a millimetre pressure are indicated by the lower curve in fig. 8 for purposes of comparison. In hydrogen at a millimetre pressure the mean free path / of an electron has a minimum value of ‘0195 cm. when the velocity wis 7X 107 em. per sec. As w diminishes / increases, and when u=13x10', 1=-044. In nitrogen a minimum -alue of J equal to *026 occurs when u=9 x 10’, and / increases to ‘057 when w=2°5 x 10%. It appears from the curves showing / in terms of u that the free paths in hydrogen and nitrogen would continue to increase with further reduction in the velocity, and it is probable that in these gases | attains a maximum value oY. 2 FFT REIS 1052 Does an Accelerated Electron radiate Energy 2 for certain velocities smaller than that corresponding to 2 volt. The large increases of the free paths of electrons as the velocity diminishes are the most remarkable of the definite results obtained from these experiments. There can be no doubt that these conclusions about the mean free paths, and the estimates of the loss of energy of the electrons in collisions with molecules, are substantially correct, notwith- standing the possible experimental errors or any uncertainty as to the exact values of the numerical coefficients in the formule that have been used. XOV. Does an Accelerated Electron necessarily radiate Energy on the Classical Theory? To the Editors of the Philosophical Magazine. GENTLEMEN,— Y the kindness of Professor Born I have learnt that the absence of radiation from the system of two oppositely charged point electrons of Lorentz mass accelerated by a uniform electric field, which I proved in a paper with this title in your March 1921 number (p. 405), also follows from a general theory which he worked out so long ago as 1999. Professor Born’s paper (Ann. d. Phys. xxx. p. 1, 1909) forms a discussion of the theory of rigidity and of the motion of a “rigid” electron, on the basis of the principle of relativity, and one of his conclusions is given in the following words :—- ‘“‘ Bemerkenswerth ist, dass ein Hlektron bei einer Hyper- belbewegung, so gross auch ihre Beschleunigung sein mag, keine eigentliche Strahlung veranlasst, sondern sein Feld mit sich fiihrt, was bis jetzt nur fiir gleichformig bewegte Elektronen bekannt war. Die Strahlung und der Widerstand der Strahlung treten erst bei Abweichungen von der Hyper- belbewegung auf.”. | This remarkable result of the early days of relativity seems to be but little known in this country, may I therefore. be permitted to direct attention to it here? By “ Hyper- belbewegung”’ is meant the motion of a particle whose world- line in the four-dimensional universe is hyperbolic, or, which comes to the same thing, the graph of which on an a, ¢ diagram forms an hyperbola. It is the equivalent in the relativity theory of uniform acceleration in Newtonian dynamics. Yours faithfully, September 30th, 1922. S. R. Mitner. iP 10530 '| XCVI. Simple Model to illustrate Elastic Hysteresis. To the Editors of the Philosophical Magazine. GENTLEMEN,-— EGARDING the very interesting paper by Mr. 8. Lees, ~ M.A., St. John’s College, Cambridge, ‘‘ On a Simple Model to illustrate Elastic Hysteresis ?’ in your current issue for September, 1922, may I be permitted to refer to an earlier publication by your present correspondent in the pages of the ‘ Physical Review’ (Ithaca, N.Y., U.S.A.) for June last. Before the Physical Society there had been described the model illustrated below, which involved both spring and la AAA A A OD Af A 6 ola memos eae oo eo a ee Ae, e, TV VU / / ev, , VV VW & AS era AR a an Aen f'n Pa See ee ee ¥, 7 ) U U UV U UJ =a — cm : t | is dip rims 3 ig 8 tin: dt dt} d\_§ aoe d{_} ‘ims H+ a8: solid frictional ‘constraints, similar to those employed by Mr. Lees. The model enabled the writer to explain not only mechani- cal hysteresis, but electrical and magnetic hysteretic systems, 1054 Simple Model to illustrate Elastic Hysteresis. involving formulze of the type B=p.H+1. The hysteresis was shown to be due, quite naturally, to the I-Component. 319 Dorset Avenue, T am, Gentlemen, Chevey Chase, Md., U.S.A. Yours etc., September 23,1922. A. PRuss. > Lo the Editors of the Philosophical Magazine. GENTLEMEN,— From the sketch given by Mr. Press, it is quite clear that his mode! will produce effects analogous to those of the model described in my paper (Phil. Mag. September 1922). Until to-day, I had not seen the short abstract of Mr. Press’s paper in the July (1922) number of the ‘ Physical Review.” I hope his full paper will be available before long. As evidence of the curious coincidence of the papers, I note that Mr. Press’s paper was read to the American Physical Society on April 21st last. My own MS. was in the hands of the publishers about April 5th last (the interval between this date and publication I am not responsible for). Referring to my fig. 3, since the model is symmetrical - about OO,, only one-half of the model need be considered in actual practice. I should like to emphasize a result arising out of my discussion, namely the fact that with either Mr. Press’s or my own suggested model, an alteration by overstrain of the point of departure of the stress-strain curve in tension from the straight-line law, is accompanied by an alteration of the corresponding point of departure in compression. This result, although of considerable technical importance, does not appear to have received much theoretical attention. 51 Chesterton Road, Yours etc., Cambridge. S. LEzs. October 6th, 1922. | [- 1055 ] XCVII. Notices respecting New Books. Cours de physique générale. Par H. Ounivirr. (Paris: Librairie Scientifique. J. Hermann, 1921.) Pus tome premier is devoted to a discussion of units, gravi- tation, electro-magnetic, in the C.G.8. system, and a new M.T:S. system (metre-tonne-second) recommended here. What is the matter with Physics Training for Engineers ? This is the question being asked to-day. The answer is sure to be: [tis the C.G.S. source of Arrogance. These niggling microscopic units are thrown aside by the young engineer as soon as he is tree from the tyranny of the lecture and examination room. They are described by Halsey in his ‘ Handbook for Draftsmen as a ‘Monument of scientific zeal with ignorance of practical requirements. The object of Weights and Measures is to Weigh and Measure, not merely to make calculations.” The second is acclaimed as the unit of time, because it keeps g down to an easily remembered number for calculations. But the engineer prefers the minute to record revolutions, as an interval that can be checked with accuracy on a stop-watch. And in the astronomical units of Relativity the unit of time is nearly 1000 years. But when it came to the choice of such diminutive units as the centimetre and gramme, in preference to the commercial metre and kilogramme, the decision was made for the supposed con- venience of making the density of water unity, so that density and specific gravity would be the same number, and a name would be saved. On the M.K.S. (metre-kilogramme system) the density of water is 1000 (kg/m*), with the advantage of keeping the allow- ance for buoyancy of the air in view, as a correction of the last figure, say a deduction of 1°25; and soa Table of Density would, if absolute, require the same deduction of 1°25 to give apparent density in air. But this correction is out of sight in the C.G.S. system, and we never hear it mentioned, although an accurate measurement must be carried out by a human being in an atmo- sphere where he can breathe, and not in vacuo. The same theoretical pedantry has influenced our author in his selection of the M.T.S. system, with the same view of keeping specific gravity and density the same figure, to the same decimal, but usually ignoring the decimal when air buoyancy makes itselt felt ; paramount in the balloon. A writer on Hydrostatics is equally loose when he tells us to neglect the pressure of the atmosphere. He should observe the ? 1056 =~—— Notices respecting New Books. distinction between pressure, as gauge or absolute, as he would be compelled on changing from the non-condensing locomotive to the condensing marine engine, or in any thermodynamical caleu- lations. In the high pressures of Internal Ballistics of Artillery, the difference may be disregarded as unimportant. But it is strange to read in an elementary text-book of this neglect, when the author is employing his favourite absolute dynamical units in Hydrostatics, and speaks of an atmospheric pressure of about seventy thousand poundals on the square foot as something of triflmg account. . On the M.T.S. system with g=9°81, m/s® the absolute unit of force would be about the heft of 10-2 kilogrammes, and an atmo- sphere of one kg/em*, or 10t/m* would be expressed by 981 in M.K.S. units. . . Our artillerist reckons his pressure in the ton/inch’, of about 150 atmospheres.